diff --git a/changes-set.txt b/changes-set.txt
index 6a8a8d09d0..b8e490cccf 100644
--- a/changes-set.txt
+++ b/changes-set.txt
@@ -85,6 +85,14 @@ make a github issue.)
DONE:
Date Old New Notes
+15-Feb-25 abeq2f eqabf
+15-Feb-25 rabeq2i reqabi
+15-Feb-25 abeq1i eqabci
+15-Feb-25 abeq2i eqabi
+15-Feb-25 abeq2d eqabd
+15-Feb-25 abeq2w eqabw
+15-Feb-25 abeq1 eqabc Commuted form of eqab
+15-Feb-25 abeq2 eqab Basic closed form
14-Feb-25 --- --- Moved surreal negation theorems
from SF's mathbox to main set.mm
10-Feb-25 --- --- Moved surreal addition theorems
diff --git a/discouraged b/discouraged
index c4b10c5e31..421af8852b 100644
--- a/discouraged
+++ b/discouraged
@@ -14414,7 +14414,6 @@ New usage of "aaanOLD" is discouraged (0 uses).
New usage of "ab0ALT" is discouraged (0 uses).
New usage of "ab0OLD" is discouraged (0 uses).
New usage of "ab0orvALT" is discouraged (0 uses).
-New usage of "abeq2OLD" is discouraged (0 uses).
New usage of "abfOLD" is discouraged (0 uses).
New usage of "ablo32" is discouraged (4 uses).
New usage of "ablo4" is discouraged (3 uses).
@@ -16587,6 +16586,7 @@ New usage of "epweonALT" is discouraged (0 uses).
New usage of "eq0ALT" is discouraged (0 uses).
New usage of "eq0OLDOLD" is discouraged (0 uses).
New usage of "eq0rdvALT" is discouraged (0 uses).
+New usage of "eqabOLD" is discouraged (0 uses).
New usage of "eqeq12OLD" is discouraged (1 uses).
New usage of "eqeq12dOLD" is discouraged (1 uses).
New usage of "eqeq1dALT" is discouraged (0 uses).
@@ -19721,7 +19721,6 @@ Proof modification of "aaanOLD" is discouraged (38 steps).
Proof modification of "ab0ALT" is discouraged (33 steps).
Proof modification of "ab0OLD" is discouraged (146 steps).
Proof modification of "ab0orvALT" is discouraged (19 steps).
-Proof modification of "abeq2OLD" is discouraged (47 steps).
Proof modification of "abfOLD" is discouraged (13 steps).
Proof modification of "abn0OLD" is discouraged (28 steps).
Proof modification of "abrexexgOLD" is discouraged (43 steps).
@@ -20486,6 +20485,7 @@ Proof modification of "epweonALT" is discouraged (86 steps).
Proof modification of "eq0ALT" is discouraged (31 steps).
Proof modification of "eq0OLDOLD" is discouraged (6 steps).
Proof modification of "eq0rdvALT" is discouraged (25 steps).
+Proof modification of "eqabOLD" is discouraged (47 steps).
Proof modification of "eqeq12OLD" is discouraged (24 steps).
Proof modification of "eqeq12dOLD" is discouraged (22 steps).
Proof modification of "eqeq1dALT" is discouraged (62 steps).
diff --git a/mmset.raw.html b/mmset.raw.html
index 1815b10320..830327ecd1 100644
--- a/mmset.raw.html
+++ b/mmset.raw.html
@@ -2022,7 +2022,7 @@
may be the same) as long as they don't conflict with any distinct variable
requirements imposed on ` A ` and ` B ` . We then would
follow the procedure of the above example. The description of theorem
-~ abeq2 goes into more detail and gives some
+~ eqab goes into more detail and gives some
actual database examples where this translation takes place.
@@ -2229,7 +2229,7 @@
Commutative law for union ~ uncom
A basic relationship between class and wff
-variables ~ abeq2
+variables ~ eqab
Two ways of saying "is a set" ~ isset
diff --git a/nf.mm b/nf.mm
index 38e95ec40a..e61dc9f55e 100644
--- a/nf.mm
+++ b/nf.mm
@@ -20257,7 +20257,7 @@ of variable (class variable) that can substituted with expressions such as
a setvar variable ` x ` for a class variable: all sets are classes by
~ cvjust (but not necessarily vice-versa). For a full description of how
classes are introduced and how to recover the primitive language, see the
- discussion in Quine (and under ~ abeq2 for a quick overview).
+ discussion in Quine (and under ~ eqab for a quick overview).
Because class variables can be substituted with compound expressions and
setvar variables cannot, it is often useful to convert a theorem
@@ -20343,7 +20343,7 @@ of logic but rather presupposes the Axiom of Extensionality (see Theorem
definition. This also makes some proofs shorter and probably easier to
read, without the constant switching between two kinds of equality.
- See also comments under ~ df-clab , ~ df-clel , and ~ abeq2 .
+ See also comments under ~ df-clab , ~ df-clel , and ~ eqab .
In the form of ~ dfcleq , this is called the "axiom of extensionality"
by [Levy] p. 338, who treats the theory of classes as an extralogical
@@ -21367,7 +21367,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
equivalent formulation cplem2 in set.mm. For more information on class
variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.
(Contributed by NM, 5-Aug-1993.) $)
- abeq2 $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
+ eqab $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
( vy cab wceq cv wcel wb wal ax-17 hbab1 cleqh abid bibi2i albii bitri )
CABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
$}
@@ -21376,8 +21376,8 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$d x A $.
$( Equality of a class variable and a class abstraction. (Contributed by
NM, 20-Aug-1993.) $)
- abeq1 $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
- ( cab wceq cv wcel wb wal abeq2 eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
+ eqabc $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
+ ( cab wceq cv wcel wb wal eqab eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
$}
@@ -21385,7 +21385,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
abeqi.1 $e |- A = { x | ph } $.
$( Equality of a class variable and a class abstraction (inference rule).
(Contributed by NM, 3-Apr-1996.) $)
- abeq2i $p |- ( x e. A <-> ph ) $=
+ eqabi $p |- ( x e. A <-> ph ) $=
( cv wcel cab eleq2i abid bitri ) BEZCFKABGZFACLKDHABIJ $.
$}
@@ -21393,7 +21393,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
abeqri.1 $e |- { x | ph } = A $.
$( Equality of a class variable and a class abstraction (inference rule).
(Contributed by NM, 31-Jul-1994.) $)
- abeq1i $p |- ( ph <-> x e. A ) $=
+ eqabci $p |- ( ph <-> x e. A ) $=
( cv cab wcel abid eleq2i bitr3i ) ABEZABFZGKCGABHLCKDIJ $.
$}
@@ -21401,7 +21401,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
abeqd.1 $e |- ( ph -> A = { x | ps } ) $.
$( Equality of a class variable and a class abstraction (deduction).
(Contributed by NM, 16-Nov-1995.) $)
- abeq2d $p |- ( ph -> ( x e. A <-> ps ) ) $=
+ eqabd $p |- ( ph -> ( x e. A <-> ps ) ) $=
( cv wcel cab eleq2d abid syl6bb ) ACFZDGLBCHZGBADMLEIBCJK $.
$}
@@ -21421,7 +21421,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$( Equality of a class variable and a class abstraction (inference rule).
(Contributed by NM, 5-Aug-1993.) $)
abbi2i $p |- A = { x | ph } $=
- ( cab wceq cv wcel wb abeq2 mpgbir ) CABEFBGCHAIBABCJDK $.
+ ( cab wceq cv wcel wb eqab mpgbir ) CABEFBGCHAIBABCJDK $.
$}
${
@@ -21460,7 +21460,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$( Deduction from a wff to a class abstraction. (Contributed by NM,
9-Jul-1994.) $)
abbi2dv $p |- ( ph -> A = { x | ps } ) $=
- ( cv wcel wb wal cab wceq alrimiv abeq2 sylibr ) ACFDGBHZCIDBCJKAOCELBCDM
+ ( cv wcel wb wal cab wceq alrimiv eqab sylibr ) ACFDGBHZCIDBCJKAOCELBCDM
N $.
$}
@@ -21470,7 +21470,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$( Deduction from a wff to a class abstraction. (Contributed by NM,
9-Jul-1994.) $)
abbi1dv $p |- ( ph -> { x | ps } = A ) $=
- ( cv wcel wb wal cab wceq alrimiv abeq1 sylibr ) ABCFDGHZCIBCJDKAOCELBCDM
+ ( cv wcel wb wal cab wceq alrimiv eqabc sylibr ) ABCFDGHZCIBCJDKAOCELBCDM
N $.
$}
@@ -21521,7 +21521,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
17-Jan-2006.) $)
clabel $p |- ( { x | ph } e. A <->
E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) ) $=
- ( cab wcel cv wceq wa wex wb wal df-clel abeq2 anbi2ci exbii bitri ) ABEZ
+ ( cab wcel cv wceq wa wex wb wal df-clel eqab anbi2ci exbii bitri ) ABEZ
DFCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
$}
@@ -21873,7 +21873,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
(Contributed by NM, 27-Sep-2003.) (Revised by Mario Carneiro,
7-Oct-2016.) $)
sbabel $p |- ( [ y / x ] { z | ph } e. A <-> { z | [ y / x ] ph } e. A ) $=
- ( vv cv cab wceq wcel wa wex wsb wb wal sbf abeq2 sbbii 3bitr4i sbex sban
+ ( vv cv cab wceq wcel wa wex wsb wb wal sbf eqab sbbii 3bitr4i sbex sban
nfv sbrbis sbalv nfcri anbi12i bitri exbii df-clel ) GHZADIZJZUKEKZLZGMZB
CNZUKABCNZDIZJZUNLZGMZULEKZBCNUSEKUQUOBCNZGMVBUOGBCUAVDVAGVDUMBCNZUNBCNZL
VAUMUNBCUBVEUTVFUNDHUKKZAOZDPZBCNVGUROZDPVEUTVHVJBCDVGVGABCVGBCVGBUCQUDUE
@@ -24063,14 +24063,14 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$( An "identity" law of concretion for restricted abstraction. Special case
of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $)
rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $=
- ( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $.
+ ( cv wcel wa crab df-rab eqabi ) BDCEAFBABCGABCHI $.
${
$d x A $.
$( An "identity" law for restricted class abstraction. (Contributed by NM,
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) $)
rabid2 $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
- ( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i
+ ( cv wcel wa cab wceq wi wal crab eqab pm4.71 albii bitr4i df-rab eqeq2i
wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
$}
@@ -24633,7 +24633,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
rabeqi.1 $e |- A = { x e. B | ph } $.
$( Inference rule from equality of a class variable and a restricted class
abstraction. (Contributed by NM, 16-Feb-2004.) $)
- rabeq2i $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
+ reqabi $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
$}
@@ -24694,7 +24694,7 @@ atomic formula (class version of ~ elsb2 ). (Contributed by Jim
$( All setvar variables are sets (see ~ isset ). Theorem 6.8 of [Quine]
p. 43. (Contributed by NM, 5-Aug-1993.) $)
vex $p |- x e. _V $=
- ( cv cvv wcel wceq eqid df-v abeq2i mpbir ) ABZCDJJEZJFKACAGHI $.
+ ( cv cvv wcel wceq eqid df-v eqabi mpbir ) ABZCDJJEZJFKACAGHI $.
${
$d x A $.
@@ -26843,7 +26843,7 @@ equiconsistent to Z (ZF in which ax-sep in set.mm replaces ax-rep in
(Contributed by NM, 7-Aug-1994.) $)
ru $p |- { x | x e/ x } e/ _V $=
( vy cv wnel cab cvv wcel wn wceq wex wb wal pm5.19 df-nel eleq12d notbid
- eleq1 id syl5bb mtbir bibi12d spv mto abeq2 nex isset mpbir ) ACZUHDZAEZF
+ eleq1 id syl5bb mtbir bibi12d spv mto eqab nex isset mpbir ) ACZUHDZAEZF
DUJFGZHUKBCZUJIZBJUMBUMUHULGZUIKZALZUPULULGZUQHZKZUQMUOUSABUHULIZUNUQUIUR
UHULULQUIUHUHGZHUTURUHUHNUTVAUQUTUHULUHULUTRZVBOPSUAUBUCUIAULUDTUEBUJUFTU
JFNUG $.
@@ -27662,7 +27662,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
NM, 5-Nov-2005.) $)
sbcabel $p |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <->
{ y | [. A / x ]. ph } e. B ) ) $=
- ( vw wcel cvv cab wsbc wb cv wceq wa wex wal abeq2 bitrd elex sbcexg sbcg
+ ( vw wcel cvv cab wsbc wb cv wceq wa wex wal eqab bitrd elex sbcexg sbcg
sbcang sbcbii sbcalg sbcbig bibi1d albidv syl6bbr anbi12d df-clel 3bitr4g
syl5bb nfcri sbcgf exbidv syl ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAUSHNZUT
OZVFEIZPZHQZBDLZVFVDOZVHPZHQZVBVEUSVKVIBDLZHQVNVIHBDJUBUSVOVMHUSVOVGBDLZV
@@ -27874,7 +27874,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
$( Composition law for chained substitutions into a class. (Contributed by
NM, 10-Nov-2005.) $)
csbco $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
- ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbcco bitri abbii 3eqtr4i
+ ( vz cv csb wcel wsbc cab df-csb eqabi sbcbii sbcco bitri abbii 3eqtr4i
) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
UCABCNOPBECTKAECDKQ $.
$}
@@ -28283,7 +28283,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.) $)
csbnestgf $p |- ( ( A e. V /\ A. y F/_ x C ) ->
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
- ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
+ ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb eqabi wb nfcr
sbcbii wnf alimi sbcnestgf sylan2 syl5bb abbidv sylan 3eqtr4g ) CFHZAEIZB
JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
@@ -32674,7 +32674,7 @@ fewer connectives (but probably less intuitive to understand).
$( Subclass relationship for power class. (Contributed by NM,
21-Jun-2009.) $)
pwss $p |- ( ~P A C_ B <-> A. x ( x C_ A -> x e. B ) ) $=
- ( cpw wss cv wcel wi wal dfss2 df-pw abeq2i imbi1i albii bitri ) BDZCEAFZ
+ ( cpw wss cv wcel wi wal dfss2 df-pw eqabi imbi1i albii bitri ) BDZCEAFZ
PGZQCGZHZAIQBEZSHZAIAPCJTUBARUASUAAPABKLMNO $.
$}
@@ -32756,7 +32756,7 @@ fewer connectives (but probably less intuitive to understand).
$( There is only one element in a singleton. Exercise 2 of [TakeutiZaring]
p. 15. (Contributed by NM, 5-Aug-1993.) $)
elsn $p |- ( x e. { A } <-> x = A ) $=
- ( cv wceq csn df-sn abeq2i ) ACBDABEABFG $.
+ ( cv wceq csn df-sn eqabi ) ACBDABEABFG $.
$}
${
@@ -33128,7 +33128,7 @@ fewer connectives (but probably less intuitive to understand).
abstraction is a singleton. (Contributed by Mario Carneiro,
14-Nov-2016.) $)
euabsn2 $p |- ( E! x ph <-> E. y { x | ph } = { y } ) $=
- ( weu cv wceq wb wal wex cab csn df-eu wcel abeq1 elsn bibi2i albii bitri
+ ( weu cv wceq wb wal wex cab csn df-eu wcel eqabc elsn bibi2i albii bitri
exbii bitr4i ) ABDABEZCEZFZGZBHZCIABJUBKZFZCIABCLUGUECUGAUAUFMZGZBHUEABUF
NUIUDBUHUCABUBOPQRST $.
@@ -33769,7 +33769,7 @@ fewer connectives (but probably less intuitive to understand).
$( The singleton of a class is a subset of its power class. (Contributed
by NM, 5-Aug-1993.) $)
snsspw $p |- { A } C_ ~P A $=
- ( vx csn cpw cv wceq wss wcel eqimss elsn df-pw abeq2i 3imtr4i ssriv ) BA
+ ( vx csn cpw cv wceq wss wcel eqimss elsn df-pw eqabi 3imtr4i ssriv ) BA
CZADZBEZAFQAGZQOHQPHQAIBAJRBPBAKLMN $.
$}
@@ -36315,7 +36315,7 @@ Kuratowski ordered pairs (continued)
$( Cardinal one is a set. (Contributed by SF, 12-Jan-2015.) $)
1cex $p |- 1c e. _V $=
( vy vx vw vz c1c cvv wcel wel weq wb wal wex ax-1c cv isset bitri bibi2i
- wceq albii exbii csn cab df-1c eqeq2i abeq2 dfcleq df-sn abeq2i mpbir ) E
+ wceq albii exbii csn cab df-1c eqeq2i eqab dfcleq df-sn eqabi mpbir ) E
FGZABHZCAHZCDIZJZCKZDLZJZAKZBLZBADCMUJBNZERZBLUSBEOVAURBVAUKANZDNZUAZRZDL
ZJZAKZURVAUTVFAUBZRVHEVIUTADUCUDVFAUTUEPVGUQAVFUPUKVEUODVEULCNVDGZJZCKUOC
VBVDUFVKUNCVJUMULUMCVDCVCUGUHQSPTQSPTPUI $.
@@ -37382,7 +37382,7 @@ Kuratowski ordered pairs (continued)
elp6 $p |- ( A e. V -> ( A e. P6 B <-> A. x << x , { A } >> e. B ) ) $=
( vy wcel cp6 cvv csn cxpk wss cv copk wal wceq sneq wi vex albii bitri
sneqd xpkeq2d sseq1d df-p6 elab2g xpkssvvk ssrelk ax-mp opkelxpk biantrur
- wb wa df-sn abeq2i 3bitr2i imbi1i snex opkeq2 eleq1d ceqsalv syl6bb ) BDF
+ wb wa df-sn eqabi 3bitr2i imbi1i snex opkeq2 eleq1d ceqsalv syl6bb ) BDF
BCGZFHBIZIZJZCKZALZVCMZCFZANZHELZIZIZJZCKVFEBVBDVKBOZVNVECVOVMVDHVOVLVCVK
BPUAUBUCECUDUEVFVGVKMZVEFZVPCFZQZENZANZVJVEHHJKVFWAUKHVDUFAEVECUGUHVTVIAV
TVKVCOZVRQZENVIVSWCEVQWBVRVQVGHFZVKVDFZULWEWBVGVKHVDARZERUIWDWEWFUJWBEVDE
@@ -38092,7 +38092,7 @@ Kuratowski ordered pairs (continued)
SF, 20-Jan-2015.) $)
unipw1 $p |- U. ~P1 A = A $=
( vx vy vz cpw1 cuni cv wcel wel wa wex csn wceq eluni elpw1 anbi1i ancom
- wrex weq 3bitri r19.41v 3bitr4i exbii risset ceqsexv abeq2i equcom rexbii
+ wrex weq 3bitri r19.41v 3bitr4i exbii risset ceqsexv eqabi equcom rexbii
snex eleq2 df-sn rexcom4 3bitr2ri eqriv ) BAEZFZABGZUPHBCIZCGZUOHZJZCKUSD
GZLZMZURJZDARZCKZUQAHZCUQUONVAVFCUTURJVDDARZURJVAVFUTVIURDUSAOPURUTQVDURD
AUAUBUCVHDBSZDARVECKZDARVGDUQAUDVKVJDAVKUQVCHZBDSZVJURVLCVCVBUIUSVCUQUJUE
@@ -39754,7 +39754,7 @@ Definite description binder (inverted iota)
wn df-rex 3bitr4i opkeq1 eleq1d ceqsexv elpw131c 19.41v excom wb elpw161c
wel elsymdif otkelins3k vex elssetk otkelins2k elpw181c ndisjrelk elcompl
bitri notbii df-ne con2bii elpw1111c orbi12i elun bibi12i wal dfcleq alex
- wo anbi12i rexcom df-addc eqeq2i abeq2 opkelimagek dfaddc2 addcex addceq1
+ wo anbi12i rexcom df-addc eqeq2i eqab opkelimagek dfaddc2 addcex addceq1
eqeq2d rexbii opkelxpk mpbiran2 elsnc anbi2i syl6bb pm5.32i 2exbii bitr3i
eldif ancom abbi2i eqtr4i vvex xpkex ssetkex ins3kex ins2kex pw1ex imakex
inex complex symdifex 1cex unex addcexlem imagekex nncex difex eqeltri )
@@ -40076,7 +40076,7 @@ Definite description binder (inverted iota)
cins2k csik cins3k csymdif c1c cimak wss ccompl cpw opkex elimak elpw121c
anbi1i 19.41v bitr4i exbii df-rex excom 3bitr4i opkeq1 ceqsexv otkelins2k
eleq1d elsymdif vex elssetk bitri otkelins3k opksnelsik opkelssetkg mp2an
- wrex bibi12i notbii elcompl cab wal df-pw eqeq2i abeq2 alex ) AGZBHZIUAZI
+ wrex bibi12i notbii elcompl cab wal df-pw eqeq2i eqab alex ) AGZBHZIUAZI
UBZUCZUDZUEJJZUFZKZLEMZBKZWKAUGZNZLZEOZLZWCWIUHKBAUIZPZWJWPWJFMZWCHZWGKZF
WHVLZWTWKGZGZGZPZXBQZFOZEOZWPFWGWHWCWBBUJZUKWTWHKZXBQZFOXHEOZFOXCXJXMXNFX
MXGEOZXBQXNXLXOXBEWTULUMXGXBEUNUOUPXBFWHUQXHEFURUSXIWOEXIXFWCHZWGKZXPWDKZ
@@ -40989,7 +40989,7 @@ Definite description binder (inverted iota)
wral elin wel wal opkex elimak elpw121c anbi1i 19.41v bitr4i df-rex excom
wb opkeq1 ceqsexv elsymdif otkelins3k elssetk bitri otkelins2k opksnelsik
eleq1d elpw141c ndisjrelk notbii elcompl con2bii elpw171c orbi12i bibi12i
- df-ne elun dfcleq alex anbi12i rexcom df-addc eqeq2i abeq2 opkelxpk elsnc
+ df-ne elun dfcleq alex anbi12i rexcom df-addc eqeq2i eqab opkelxpk elsnc
weq elpw11c opkelcnvk opkelimagek dfaddc2 addcex addceq1 eqeq2d opkelidkg
mpbiran2 eldif mp2an equcom 1cex eqeq2 eqeq1 notbid anbi12d annim ssetkex
rexbii ins3kex ins2kex inex pw1ex imakex complex sikex unex symdifex vvex
@@ -41210,7 +41210,7 @@ Definite description binder (inverted iota)
cab cpw1 cxpk cin ccompl cins3k cins2k cv cun csymdif csik cdif cuni wceq
cimak wrex wral snex opkeq1 eleq1d ceqsexv elin vex elssetk eldif elcompl
wi elimak el1c anbi1i 19.41v bitr4i df-rex excom opkelxpk mpbiran2 elequ2
- snelpw1 elab anbi12i eluni xchbinx wb wal abeq2 opkex elpw121c otkelins3k
+ snelpw1 elab anbi12i eluni xchbinx wb wal eqab opkex elpw121c otkelins3k
df-clel opksnelsik elsymdif bitri otkelins2k alex dfcleq wo elsnc orbi12i
sneqb elun bibi12i 3bitr4ri notbii annim dfral2 abbi2i ssetkex setswithex
weq pw1ex vvex xpkex inex 1cex imakex complex ins3kex unex symdifex sikex
@@ -41460,7 +41460,7 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \
cin wsfin opkelxpk wrex opkex elimak elpw131c anbi1i 19.41v bitr4i df-rex
excom opkeq1 eleq1d ceqsexv elin elpw121c otkelins3k opksnelsik eqpw1relk
w3a vex otkelins2k elssetk bitri anbi12i df-clel wel elsymdif opkelssetkg
- wss wb wal wn mp2an bibi12i notbii elcompl alex df-pw eqeq2i abeq2 df-3an
+ wss wb wal wn mp2an bibi12i notbii elcompl alex df-pw eqeq2i eqab df-3an
cab df-sfin ) ABIZJJUAZKZWSUBUCUDUALUEZLUFZUGZUHZUBUIUIZUIZUJUKZUFZUEZLUG
ZUNZXFUJZUEZXEXFUJZULZUFZUEZXKUNZXFUJZUGZUNZXGUJZKZMZAJKZBJKZEUMZUIZAKZYH
UCZBKZMZENZVNZWSWTYCUNKABUOYEYFYGMZYNMYOXAYPYDYNABJJCDUPYDFUMZWSIZYBKZFXG
@@ -41689,7 +41689,7 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \
wral wel eleq1d ceqsexv elin opksnelsik elssetk opkelxpk mpbiran2 snelpw1
eldif otkelins2k opkelcnvk eqtfinrelk opkex elimak elpw121c anbi1i 19.41v
bitr4i df-rex excom wb elsymdif otkelins3k anbi12i bibi12i notbii elcompl
- elpw abeq2 df-clel elpw11c tfinex clel3 annim dfral2 abbi2i ssetkex sikex
+ elpw eqab df-clel elpw11c tfinex clel3 annim dfral2 abbi2i ssetkex sikex
wal alex nncex pwex pw1ex vvex xpkex tfinrelkex ins2kex ins3kex inex 1cex
cnvkex imakex symdifex complex difex uni1ex eqeltrri ) HUAZUBUCZUDZUEUFZU
AZUGIZIZYMUFHUHZHUIUJUBUEUFYHUHZUKUCUEUFHUJZYPULUKUDZUDZUDZJUMUAUJYOUNYSJ
@@ -42743,7 +42743,7 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \
phi011lem1 $p |- ( ( Phi A u. { 0c } ) = ( Phi B u. { 0c } ) ->
Phi A C_ Phi B ) $=
( vz cphi c0c csn wceq cv wcel wn ssun1 sseli eleq2 syl5ib 0cnelphi eleq1
- cun wi mtbiri wo con2i a1i elun df-sn abeq2i orbi2i bitri biimpi ord ee22
+ cun wi mtbiri wo con2i a1i elun df-sn eqabi orbi2i bitri biimpi ord ee22
orcomd ssrdv ) ADZEFZQZBDZUNQZGZCUMUPURCHZUMIZUSUQIZUSEGZJZUSUPIZUTUSUOIU
RVAUMUOUSUMUNKLUOUQUSMNUTVCRURVBUTVBUTEUMIAOUSEUMPSUAUBVAVBVDVAVDVBVAVDVB
TZVAVDUSUNIZTVEUSUPUNUCVFVBVDVBCUNCEUDUEUFUGUHUKUIUJUL $.
@@ -42961,7 +42961,7 @@ Ins3_k SI_k ( ( ~P 1c X._k _V ) \
phidisjnn $p |- ( ( A i^i Nn ) = (/) -> Phi A = A ) $=
( vx vy cnnc cin c0 wceq cv wcel c1c cplc cif wrex wb wal cphi wa syl6bbr
weq wn wral disj biimpi r19.21bi iffalse syl eqeq2d equcom risset alrimiv
- rexbidva cab df-phi eqeq1i abeq1 bitri sylibr ) ADEFGZBHZCHZDIZUTJKZUTLZG
+ rexbidva cab df-phi eqeq1i eqabc bitri sylibr ) ADEFGZBHZCHZDIZUTJKZUTLZG
ZCAMZUSAIZNZBOZAPZAGZURVGBURVECBSZCAMVFURVDVKCAURUTAIQZVDBCSVKVLVCUTUSVLV
ATZVCUTGURVMCAURVMCAUACADUBUCUDVAVBUTUEUFUGCBUHRUKCUSAUIRUJVJVEBULZAGVHVI
VNACBAUMUNVEBAUOUPUQ $.
@@ -45412,7 +45412,7 @@ We use their notation ("onto" under the arrow). For alternate
$( Representation of a constant function using ordered pairs. (Contributed
by NM, 12-Oct-1999.) $)
fconstopab $p |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y = B ) } $=
- ( csn cxp cv wcel wa copab wceq df-xp df-sn abeq2i anbi2i opabbii eqtri )
+ ( csn cxp cv wcel wa copab wceq df-xp df-sn eqabi anbi2i opabbii eqtri )
CDEZFAGCHZBGZRHZIZABJSTDKZIZABJABCRLUBUDABUAUCSUCBRBDMNOPQ $.
$}
@@ -46413,7 +46413,7 @@ We use their notation ("onto" under the arrow). For alternate
dmopab3 $p |- ( A. x e. A E. y ph <->
dom { <. x , y >. | ( x e. A /\ ph ) } = A ) $=
( wex wral cv wcel wi wal wa wb copab cdm wceq df-ral pm4.71 albii dmopab
- cab 19.42v abbii eqtri eqeq1i eqcom abeq2 3bitr2ri 3bitri ) ACEZBDFBGDHZU
+ cab 19.42v abbii eqtri eqeq1i eqcom eqab 3bitr2ri 3bitri ) ACEZBDFBGDHZU
IIZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCE
ZBTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
$}
@@ -46449,7 +46449,7 @@ We use their notation ("onto" under the arrow). For alternate
$( An empty domain implies an empty range. (Contributed by set.mm
contributors, 21-May-1998.) $)
dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
- ( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii abeq1 3bitr4i
+ ( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii eqabc 3bitr4i
wn nbn eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i dfdm2 dfrn2 ) BDZCDZAE
ZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULRZBLZUORZC
LZVBVEVGUOCFZRVIVGULBFVJULBMUKBCUCUDUOCMUEVFVABUTULUINSOVHVDCVCUOUJNSOUFU
@@ -50336,7 +50336,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ).
E. x e. B ( F ` x ) = C ) ) $=
( vy wfn wss wa cima wcel cv cfv wceq wrex cvv anim2i eleq1 wb wi rexbidv
elex fvex mpbii rexlimivw eqeq2 bibi12d imbi2d wfun cdm fnfun adantr fndm
- cab sseq2d biimpar dfimafn syl2anc abeq2d vtoclg impcom pm5.21nd ) EBGZCB
+ cab sseq2d biimpar dfimafn syl2anc eqabd vtoclg impcom pm5.21nd ) EBGZCB
HZIZDECJZKZALZEMZDNZACOZVEDPKZIVGVLVEDVFUBQVKVLVEVJVLACVJVIPKVLVHEUCVIDPR
UDUEQVLVEVGVKSZVEFLZVFKZVIVNNZACOZSZTVEVMTFDPVNDNZVRVMVEVSVOVGVQVKVNDVFRV
SVPVJACVNDVIUFUAUGUHVEVQFVFVEEUIZCEUJZHZVFVQFUNNVCVTVDBEUKULVCWBVDVCWABCB
@@ -50889,7 +50889,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ).
A. y e. B E. x e. A y = ( F ` x ) ) ) $=
( wfo wf crn wceq wa cv cfv wrex wral dffo2 cab wb wcel wal wi wfn fnrnfv
ffn eqeq1d simpr ffvelrn adantr eqeltrd exp31 rexlimdv biantrurd syl6rbbr
- syl dfbi2 albidv abeq1 df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZD
+ syl dfbi2 albidv eqabc df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZD
IZJVBBKZAKZELZIZACMZBDNZJCDEOVBVDVJVBVDVIBPZDIZVJVBECUAZVDVLQCDEUCVMVCVKD
ABCEUBUDUMVBVIVEDRZQZBSVNVITZBSVLVJVBVOVPBVBVPVIVNTZVPJVOVBVQVPVBVHVNACVB
VFCRZVHVNVBVRJZVHJVEVGDVSVHUEVSVGDRVHCDVFEUFUGUHUIUJUKVIVNUNULUOVIBDUPVIB
@@ -54206,7 +54206,7 @@ result of an operator (deduction version). (Contributed by Paul
` A ` . (Contributed by Mario Carneiro, 13-Feb-2015.) $)
fvmptss $p |- ( A. x e. A B C_ C -> ( F ` D ) C_ C ) $=
( vy wss wcel cfv cv wi wceq fveq2 sseq1d imbi2d nfcv wa c0 dmmptss sseli
- wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv dmmpt rabeq2i fvmpt2
+ wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv dmmpt reqabi fvmpt2
eqimss syl sylbi ndmfv 0ss a1i eqsstrd pm2.61i rsp impcom syl5ss vtoclgaf
wn ex vtoclga sylan2 adantl pm2.61dan ) CDIZABUCZEFUDZJZEFKZDIZVQVOEBJZVS
VPBEABCFGUAUBVTVOVSVOHLZFKZDIZMZVOVSMHEBWAENZWCVSVOWEWBVRDWAEFOPQVOALZFKZ
@@ -55972,7 +55972,7 @@ result of an operator (deduction version). (Contributed by Paul
9-Mar-2015.) $)
fvfullfunlem3 $p |- ( A e. dom ( ( _I o. F ) \ ( ~ _I o. F ) ) ->
( ( ( _I o. F ) \ ( ~ _I o. F ) ) ` A ) = ( F ` A ) ) $=
- ( vx vy vz cid ccom wcel cfv wss wceq cv wbr wal weu fvfullfunlem1 abeq2i
+ ( vx vy vz cid ccom wcel cfv wss wceq cv wbr wal weu fvfullfunlem1 eqabi
wa brres cvv ccompl cdif cdm cres wfun weq wi dffun2 anbi2i bitri adantrl
tz6.12-1 adantl eqtr3d adantlr syl2anb gen2 cxp cin fvfullfunlem2 crn ssv
mpgbir ssdmrn xpss2 ax-mp sstri ssini df-res sseqtr4i funssfv mp3an12
@@ -55992,7 +55992,7 @@ result of an operator (deduction version). (Contributed by Paul
( vx vy cvv wcel cfv wceq cv fveq2 cid ccom ccompl c0 wfn wa mp3an12 mpan
0ex eqtr4d cfullfun eqeq12d cdm csn cxp cun df-fullfun fveq1i cin incompl
cdif wfun fnfullfunlem2 funfn fnconstg ax-mp fvun1 fvfullfunlem3 eqtrd wn
- mpbi vex elcompl sylbir wbr fvfullfunlem1 abeq2i tz6.12-2 sylnbi fvconst2
+ mpbi vex elcompl sylbir wbr fvfullfunlem1 eqabi tz6.12-2 sylnbi fvconst2
fvun2 weu pm2.61i eqtri vtoclg fvprc ) AEFZABUAZGZABGZHZCIZVRGZWBBGZHWACA
EWBAHWCVSWDVTWBAVRJWBABJUBWCWBKBLKMBLUKZWEUCZMZNUDUEZUFZGZWDWBVRWIBUGUHWB
WFFZWJWDHWKWJWBWEGZWDWFWGUINHZWKWJWLHZWFUJZWEWFOZWHWGOZWMWKPWNWEULWPBUMWE
@@ -57812,7 +57812,7 @@ the first case of his notation (simple exponentiation) and subscript it
pmvalg $p |- ( ( A e. C /\ B e. D ) ->
( A ^pm B ) = { f e. ~P ( B X. A ) | Fun f } ) $=
( vx vy wcel cvv cpm cv cxp cpw crab wceq elex wa cab pweqd biidd co wfun
- wss df-rab ancom df-pw abeq2i anbi2i bitri abbii eqtri syl5eqel rabeqbidv
+ wss df-rab ancom df-pw eqabi anbi2i bitri abbii eqtri syl5eqel rabeqbidv
pmex ancoms xpeq2 xpeq1 df-pm ovmpt2g mpd3an3 syl2an ) ACHAIHZBIHZABJUAEK
ZUBZEBALZMZNZOZBDHACPBDPVBVCVHIHVIVBVCQVHVEVDVFUCZQZERZIVHVDVGHZVEQZERVLV
EEVGUDVNVKEVNVEVMQVKVMVEUEVMVJVEVJEVGEVFUFUGUHUIUJUKVCVBVLIHBAIIEUNUOULFG
@@ -58716,7 +58716,7 @@ the first case of his notation (simple exponentiation) and subscript it
enpw1pw $p |- ~P1 ~P A ~~ ~P ~P1 A $=
( vy vx vz cpw cpw1 cpw1fn wf1o wbr c1c wss ax-mp wceq cv wrex wa wex vex
wcel cres cen cima wf1 pw1fnf1o f1of1 pw1ss1c f1ores mp2an wb df-ima elpw
- cab sspw1 df-rex df-pw abeq2i anbi1i exbii bitr2i 3bitri csn elpw1 bitr4i
+ cab sspw1 df-rex df-pw eqabi anbi1i exbii bitr2i 3bitri csn elpw1 bitr4i
r19.41v rexcom4 snex breq1 ceqsexv bitri rexbii bitr3i abbi2i eqtr4i mpbi
brpw1fn f1oeq3 pw1fnex pwex pw1ex resex f1oen ) AFZGZAGZFZHWDUAZIZWDWFUBJ
WDHWDUCZWGIZWHKKFZHUDZWDKLWJKWKHIWLUEKWKHUFMWCUGKWKWDHUHUIWIWFNWJWHUJWICO
@@ -59805,7 +59805,7 @@ the first case of his notation (simple exponentiation) and subscript it
wf wb wn snex opex elcompl elsymdif opelssetsn otelins2 otsnelsi3 wfn wss
df-br brfns bitr3i opelco brimage dfrn5 eqeq2i bitr4i brsset rnex ceqsexv
exbii sseq1 oteltxp bibi12i xchbinx exnal 3bitrri con1bii oqelins4 mapval
- df-f abeq2 ovex breq2 df-3an abbi2i cnvex imaexg mpan xpexg syl2an cnvexg
+ df-f eqab ovex breq2 df-3an abbi2i cnvex imaexg mpan xpexg syl2an cnvexg
pw1fnex ins3exg syl ssetex ins3ex fnsex 2ndex coex ins2ex 1cex mpan2 3syl
imageex txpex si3ex symdifex imaex complex ins4ex enex inexg syl5eqelr
inex ) CDJZBEJZKZFUAZUBZCJZGUAZUBZBJZAUAZUUNUUQUEUCZLMZUDZGNZFNZAUFOUGZCU
diff --git a/set.mm b/set.mm
index 9badafe124..78158669ae 100644
--- a/set.mm
+++ b/set.mm
@@ -24784,7 +24784,7 @@ yield an eliminable and weakly (that is, object-level) conservative
theory axioms. He calls the construction ` { y | ph } ` a "class term".
For a full description of how classes are introduced and how to recover
the primitive language, see the books of Quine and Levy (and the comment
- of ~ abeq2 for a quick overview). For a general discussion of the theory
+ of ~ eqab for a quick overview). For a general discussion of the theory
of classes, see ~ mmset.html#class . (Contributed by NM, 26-May-1993.)
(Revised by BJ, 19-Aug-2023.) $)
df-clab $a |- ( x e. { y | ph } <-> [ x / y ] ph ) $.
@@ -25221,7 +25221,7 @@ variables at the beginning of this section (not visible on the webpages).
it does not have the usual form of a definition. If we required a
definition to have the usual form, we would call ~ df-cleq an axiom.
- See also comments under ~ df-clab , ~ df-clel , and ~ abeq2 .
+ See also comments under ~ df-clab , ~ df-clel , and ~ eqab .
In the form of ~ dfcleq , this is called the "axiom of extensionality"
by [Levy] p. 338, who treats the theory of classes as an extralogical
@@ -26019,13 +26019,13 @@ the definition of class equality ( ~ df-cleq ). Its forward implication
${
$d x y $. $d y A $. $d ph y $. $d ps x $.
- abeq2w.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
- $( Version of ~ abeq2 using implicit substitution, which requires fewer
+ eqabw.1 $e |- ( x = y -> ( ph <-> ps ) ) $.
+ $( Version of ~ eqab using implicit substitution, which requires fewer
axioms. (Contributed by GG and AV, 18-Sep-2024.) $)
- abeq2w $p |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) ) $=
+ eqabw $p |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) ) $=
( cab wceq cv wcel wb wal dfcleq wsb df-clab sbievw bitri bibi2i albii )
EACGZHDIZEJZUATJZKZDLUBBKZDLDETMUDUEDUCBUBUCACDNBADCOABCDFPQRSQ $.
- $( $j usage 'abeq2w' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
+ $( $j usage 'eqabw' avoids 'df-clel' 'ax-8' 'ax-10' 'ax-11' 'ax-12'; $)
$}
@@ -26053,7 +26053,7 @@ the definition of class equality ( ~ df-cleq ). Its forward implication
it does not have the usual form of a definition. If we required a
definition to have the usual form, we would call ~ df-clel an axiom.
- See also comments under ~ df-clab , ~ df-cleq , and ~ abeq2 .
+ See also comments under ~ df-clab , ~ df-cleq , and ~ eqab .
Alternate characterizations of ` A e. B ` when either ` A ` or ` B ` is
a set are given by ~ clel2g , ~ clel3g , and ~ clel4g .
@@ -26688,6 +26688,13 @@ generally appear in a single form (either definitional, but more often
${
$d x A y $. $d ph y $.
+ $( One direction of ~ eqab is provable from fewer axioms. (Contributed by
+ Wolf Lammen, 13-Feb-2025.) $)
+ eqabr $p |- ( A. x ( x e. A <-> ph ) -> A = { x | ph } ) $=
+ ( cv wcel wb wal cab wceq abbi1 abid1 eqeq1i sylibr ) BDCEZAFBGNBHZABHZIC
+ PINABJCOPBCKLM $.
+ $( $j usage 'eqabr' avoids 'ax-10' 'ax-11' 'ax-12'; $)
+
$( Equality of a class variable and a class abstraction (also called a
class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the
relationship between expressions with class abstractions and expressions
@@ -26712,60 +26719,54 @@ generally appear in a single form (either definitional, but more often
equivalent formulation ~ cplem2 . For more information on class
variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13.
- Usage of ~ abeq2w is preferred since it requires fewer axioms.
+ Usage of ~ eqabw is preferred since it requires fewer axioms.
(Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen,
12-Feb-2025.) $)
- abeq2 $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
+ eqab $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
( cab wceq cv wcel wb wal abid1 eqeq1i abbi bitr4i ) CABDZEBFCGZBDZNEOAHB
ICPNBCJKOABLM $.
- $( Obsolete version of ~ abeq2 as of 12-Feb-2025. (Contributed by NM,
+ $( Obsolete version of ~ eqab as of 12-Feb-2025. (Contributed by NM,
26-May-1993.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
- abeq2OLD $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
+ eqabOLD $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
( vy cab wceq cv wcel wb wal ax-5 hbab1 cleqh abid bibi2i albii bitri ) C
ABEZFBGZCHZSRHZIZBJTAIZBJBDCRDGCHBKABDLMUBUCBUAATABNOPQ $.
-
- $( One direction of ~ abeq2 is provable from fewer axioms. (Contributed by
- Wolf Lammen, 13-Feb-2025.) $)
- abeq2impr $p |- ( A. x ( x e. A <-> ph ) -> A = { x | ph } ) $=
- ( cv wcel wb wal cab wceq abbi1 abid1 eqeq1i sylibr ) BDCEZAFBGNBHZABHZIC
- PINABJCOPBCKLM $.
$}
${
$d x A $.
$( Equality of a class variable and a class abstraction. Commuted form of
- ~ abeq2 . (Contributed by NM, 20-Aug-1993.) $)
- abeq1 $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
- ( cab wceq cv wcel wb wal abeq2 eqcom bicom albii 3bitr4i ) CABDZEBFCGZAH
- ZBIOCEAPHZBIABCJOCKRQBAPLMN $.
+ ~ eqab . (Contributed by NM, 20-Aug-1993.) $)
+ eqabc $p |- ( { x | ph } = A <-> A. x ( ph <-> x e. A ) ) $=
+ ( cab wceq cv wcel wb wal eqab eqcom bicom albii 3bitr4i ) CABDZEBFCGZAHZ
+ BIOCEAPHZBIABCJOCKRQBAPLMN $.
$}
${
- abeq2d.1 $e |- ( ph -> A = { x | ps } ) $.
+ eqabd.1 $e |- ( ph -> A = { x | ps } ) $.
$( Equality of a class variable and a class abstraction (deduction form of
- ~ abeq2 ). (Contributed by NM, 16-Nov-1995.) $)
- abeq2d $p |- ( ph -> ( x e. A <-> ps ) ) $=
+ ~ eqab ). (Contributed by NM, 16-Nov-1995.) $)
+ eqabd $p |- ( ph -> ( x e. A <-> ps ) ) $=
( cv wcel cab eleq2d abid bitrdi ) ACFZDGLBCHZGBADMLEIBCJK $.
$}
${
- abeq2i.1 $e |- A = { x | ph } $.
+ eqabi.1 $e |- A = { x | ph } $.
$( Equality of a class variable and a class abstraction (inference form).
(Contributed by NM, 3-Apr-1996.) (Proof shortened by Wolf Lammen,
15-Nov-2019.) $)
- abeq2i $p |- ( x e. A <-> ph ) $=
- ( cv wcel wb wtru cab wceq a1i abeq2d mptru ) BECFAGHABCCABIJHDKLM $.
+ eqabi $p |- ( x e. A <-> ph ) $=
+ ( cv wcel wb wtru cab wceq a1i eqabd mptru ) BECFAGHABCCABIJHDKLM $.
$}
${
- abeq1i.1 $e |- { x | ph } = A $.
+ eqabci.1 $e |- { x | ph } = A $.
$( Equality of a class variable and a class abstraction (inference form).
(Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen,
15-Nov-2019.) $)
- abeq1i $p |- ( ph <-> x e. A ) $=
- ( cv wcel cab eqcomi abeq2i bicomi ) BECFAABCABGCDHIJ $.
+ eqabci $p |- ( ph <-> x e. A ) $=
+ ( cv wcel cab eqcomi eqabi bicomi ) BECFAABCABGCDHIJ $.
$}
${
@@ -26798,8 +26799,8 @@ generally appear in a single form (either definitional, but more often
17-Jan-2006.) $)
clabel $p |- ( { x | ph } e. A <->
E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) ) $=
- ( cab wcel cv wceq wa wex wb wal dfclel abeq2 anbi2ci exbii bitri ) ABEZD
- FCGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
+ ( cab wcel cv wceq wa wex wb wal dfclel eqab anbi2ci exbii bitri ) ABEZDF
+ CGZRHZSDFZIZCJUABGSFAKBLZIZCJCRDMUBUDCTUCUAABSNOPQ $.
$}
${
@@ -27359,15 +27360,15 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is
$}
${
- abeq2f.0 $e |- F/_ x A $.
+ eqabf.0 $e |- F/_ x A $.
$( Equality of a class variable and a class abstraction. In this version,
the fact that ` x ` is a nonfree variable in ` A ` is explicitly stated
as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid
~ ax-13 . (Revised by Wolf Lammen, 13-May-2023.) $)
- abeq2f $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
+ eqabf $p |- ( A = { x | ph } <-> A. x ( x e. A <-> ph ) ) $=
( cab wceq cv wcel wb wal nfab1 cleqf abid bibi2i albii bitri ) CABEZFBGZ
CHZRQHZIZBJSAIZBJBCQDABKLUAUBBTASABMNOP $.
- $( $j usage 'abeq2f' avoids 'ax-13'; $)
+ $( $j usage 'eqabf' avoids 'ax-13'; $)
$}
${
@@ -31821,7 +31822,7 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is
$( An "identity" law of concretion for restricted abstraction. Special case
of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.) $)
rabid $p |- ( x e. { x e. A | ph } <-> ( x e. A /\ ph ) ) $=
- ( cv wcel wa crab df-rab abeq2i ) BDCEAFBABCGABCHI $.
+ ( cv wcel wa crab df-rab eqabi ) BDCEAFBABCGABCHI $.
$( Membership in a restricted abstraction, implication. (Contributed by
Glauco Siliprandi, 26-Jun-2021.) $)
@@ -31829,10 +31830,10 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is
( cv crab wcel rabid simplbi ) BDZABCEFICFAABCGH $.
${
- rabeq2i.1 $e |- A = { x e. B | ph } $.
+ reqabi.1 $e |- A = { x e. B | ph } $.
$( Inference from equality of a class variable and a restricted class
abstraction. (Contributed by NM, 16-Feb-2004.) $)
- rabeq2i $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
+ reqabi $p |- ( x e. A <-> ( x e. B /\ ph ) ) $=
( cv wcel crab wa eleq2i rabid bitri ) BFZCGMABDHZGMDGAICNMEJABDKL $.
$}
@@ -31892,9 +31893,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Revised
by Thierry Arnoux, 13-Mar-2017.) $)
rabid2f $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
- ( cv wcel wa cab wceq wi wal crab wral abeq2f pm4.71 bitr4i df-rab eqeq2i
- wb albii df-ral 3bitr4i ) CBECFZAGZBHZIZUCAJZBKZCABCLZIABCMUFUCUDSZBKUHUD
- BCDNUGUJBUCAOTPUIUECABCQRABCUAUB $.
+ ( cv wcel wa cab wceq wi wal crab eqabf pm4.71 albii bitr4i df-rab eqeq2i
+ wral wb df-ral 3bitr4i ) CBECFZAGZBHZIZUCAJZBKZCABCLZIABCSUFUCUDTZBKUHUDB
+ CDMUGUJBUCANOPUIUECABCQRABCUAUB $.
$}
${
@@ -31909,9 +31910,9 @@ atomic formula (class version of ~ elsb1 ). Usage of this theorem is
9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
(Proof modification is discouraged.) (New usage is discouraged.) $)
rabid2OLD $p |- ( A = { x e. A | ph } <-> A. x e. A ph ) $=
- ( cv wcel wa cab wceq wi wal crab abeq2 pm4.71 albii bitr4i df-rab eqeq2i
- wral wb df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCRUEUBUCSZBJUGUCB
- CLUFUIBUBAMNOUHUDCABCPQABCTUA $.
+ ( cv wcel wa cab wceq wi wal crab wral wb eqab pm4.71 albii bitr4i df-rab
+ eqeq2i df-ral 3bitr4i ) CBDCEZAFZBGZHZUBAIZBJZCABCKZHABCLUEUBUCMZBJUGUCBC
+ NUFUIBUBAOPQUHUDCABCRSABCTUA $.
$}
${
@@ -34316,7 +34317,7 @@ general is seen either by substitution (when the variable ` V ` has no
20-Apr-2022.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
rabeqcOLD $p |- { x e. A | ph } = A $=
- ( crab cv wcel wa cab df-rab wceq wb abeq1 pm4.71i bicomi mpgbir eqtri )
+ ( crab cv wcel wa cab df-rab wceq wb eqabc pm4.71i bicomi mpgbir eqtri )
ABCEBFCGZAHZBIZCABCJTCKSRLBSBCMRSRADNOPQ $.
$}
@@ -35513,9 +35514,9 @@ too large (beyond the size of those used in standard mathematics), the
(Proof modification is discouraged.) $)
ru $p |- { x | x e/ x } e/ _V $=
( vy cv wnel cab cvv wcel wceq wex wel wb wn pm5.19 eleq1w df-nel eleq12d
- wal id notbid mtbir bitrid bibi12d spvv mto abeq2 nex isset nelir ) ACZUI
- DZAEZFUKFGBCZUKHZBIUMBUMABJZUJKZAQZUPBBJZUQLZKZUQMUOUSABUIULHZUNUQUJURABU
- LNUJAAJZLUTURUIUIOUTVAUQUTUIULUIULUTRZVBPSUAUBUCUDUJAULUETUFBUKUGTUH $.
+ wal id notbid mtbir bitrid bibi12d spvv mto eqab nex isset nelir ) ACZUID
+ ZAEZFUKFGBCZUKHZBIUMBUMABJZUJKZAQZUPBBJZUQLZKZUQMUOUSABUIULHZUNUQUJURABUL
+ NUJAAJZLUTURUIUIOUTVAUQUTUIULUIULUTRZVBPSUAUBUCUDUJAULUETUFBUKUGTUH $.
$( $j usage 'ru' avoids 'ax-13' 'ax-reg'; $)
$}
@@ -36561,13 +36562,13 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
NM, 5-Nov-2005.) $)
sbcabel $p |- ( A e. V -> ( [. A / x ]. { y | ph } e. B <->
{ y | [. A / x ]. ph } e. B ) ) $=
- ( vw wcel cvv cab wsbc wb cv wceq wa wex wal bitrid abeq2 elex sbcan sbcg
- sbcex2 sbcal sbcbig bibi1d bitrd albidv sbcbii 3bitr4g nfcri sbcgf exbidv
- anbi12d dfclel syl ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAURHNZUSOZVEEIZPZHQ
- ZBDLZVEVCOZVGPZHQZVAVDVJVHBDLZHQURVMVHHBDUDURVNVLHVNVFBDLZVGBDLZPURVLVFVG
- BDUBURVOVKVPVGURCNVEIZAMZCRZBDLZVQVBMZCRZVOVKVTVRBDLZCRURWBVRCBDUEURWCWAC
- URWCVQBDLZVBMWAVQABDJUFURWDVQVBVQBDJUCUGUHUISVFVSBDACVETUJVBCVETUKVGBDJBH
- EGULUMUOSUNSUTVIBDHUSEUPUJHVCEUPUKUQ $.
+ ( vw wcel cvv cab wsbc wb cv wceq wa wex wal bitrid eqab elex sbcan sbcal
+ sbcex2 sbcbig sbcg bibi1d bitrd albidv sbcbii 3bitr4g nfcri sbcgf anbi12d
+ exbidv dfclel syl ) DFIDJIZACKZEIZBDLZABDLZCKZEIZMDFUAURHNZUSOZVEEIZPZHQZ
+ BDLZVEVCOZVGPZHQZVAVDVJVHBDLZHQURVMVHHBDUDURVNVLHVNVFBDLZVGBDLZPURVLVFVGB
+ DUBURVOVKVPVGURCNVEIZAMZCRZBDLZVQVBMZCRZVOVKVTVRBDLZCRURWBVRCBDUCURWCWACU
+ RWCVQBDLZVBMWAVQABDJUEURWDVQVBVQBDJUFUGUHUISVFVSBDACVETUJVBCVETUKVGBDJBHE
+ GULUMUNSUOSUTVIBDHUSEUPUJHVCEUPUKUQ $.
$}
${
@@ -36940,7 +36941,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
~ ax-13 . (Contributed by NM, 10-Nov-2005.) Avoid ~ ax-13 . (Revised
by Gino Giotto, 10-Jan-2024.) $)
csbcow $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
- ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbccow bitri abbii 3eqtr4i
+ ( vz cv csb wcel wsbc cab df-csb eqabi sbcbii sbccow bitri abbii 3eqtr4i
) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
UCABCNOPBECTKAECDKQ $.
$( $j usage 'csbcow' avoids 'ax-13'; $)
@@ -36953,9 +36954,9 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
~ csbcow when possible. (Contributed by NM, 10-Nov-2005.)
(New usage is discouraged.) $)
csbco $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $=
- ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbcco bitri abbii 3eqtr4i
- ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM
- UCABCNOPBECTKAECDKQ $.
+ ( vz cv csb wcel wsbc cab df-csb eqabi sbcbii sbcco bitri abbii 3eqtr4i )
+ EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLMUC
+ ABCNOPBECTKAECDKQ $.
$}
${
@@ -37005,8 +37006,8 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use
$( Substitution for a variable not free in a class does not affect it, in
inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) $)
csbgfi $p |- [_ A / x ]_ B = B $=
- ( vy csb cv wcel wsbc df-csb abeq2i nfcri sbcgfi bitri eqriv ) FABCGZCFHZ
- QIRCIZABJZSTFQAFBCKLSABDAFCEMNOP $.
+ ( vy csb cv wcel wsbc df-csb eqabi nfcri sbcgfi bitri eqriv ) FABCGZCFHZQ
+ IRCIZABJZSTFQAFBCKLSABDAFCEMNOP $.
$}
${
@@ -37646,9 +37647,9 @@ technically classes despite morally (and provably) being sets, like ` 1 `
8-Jan-2002.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
dfss2OLD $p |- ( A C_ B <-> A. x ( x e. A -> x e. B ) ) $=
- ( wss cv wcel wa wb wal wi cin wceq dfss df-in eqeq2i abeq2 3bitri pm4.71
- cab albii bitr4i ) BCDZAEZBFZUDUCCFZGZHZAIZUDUEJZAIUBBBCKZLBUFASZLUHBCMUJ
- UKBABCNOUFABPQUIUGAUDUERTUA $.
+ ( wss cv wcel wa wb wal cin wceq cab dfss df-in eqeq2i eqab 3bitri pm4.71
+ wi albii bitr4i ) BCDZAEZBFZUDUCCFZGZHZAIZUDUESZAIUBBBCJZKBUFALZKUHBCMUJU
+ KBABCNOUFABPQUIUGAUDUERTUA $.
$( Alternate definition of subclass relationship. (Contributed by NM,
14-Oct-1999.) $)
@@ -41233,11 +41234,11 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets,
Giotto, 26-Jan-2024.) $)
csbnestgfw $p |- ( ( A e. V /\ A. y F/_ x C ) ->
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
- ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
- sbcbii wnf alimi sbcnestgfw sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZ
- BJZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVB
- GURUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDP
- UFUGUHUIUJAGCUPSBGVAESUK $.
+ ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb eqabi wnf nfcr
+ sbcbii wb alimi sbcnestgfw sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZB
+ JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
+ URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUAZBJVGVBUDUMVHBAGEUBUEUTABCDPU
+ FUGUHUIUJAGCUPSBGVAESUK $.
$( $j usage 'csbnestgfw' avoids 'ax-13'; $)
$}
@@ -41303,11 +41304,11 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets,
Mario Carneiro, 10-Nov-2016.) (New usage is discouraged.) $)
csbnestgf $p |- ( ( A e. V /\ A. y F/_ x C ) ->
[_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C ) $=
- ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb abeq2i wb nfcr
- sbcbii wnf alimi sbcnestgf sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZB
- JZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBG
- URUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUDZBJVGVBUAUMVHBAGEUBUEUTABCDPU
- FUGUHUIUJAGCUPSBGVAESUK $.
+ ( vz wcel wnfc wal wa cv csb wsbc cab cvv wceq elex df-csb eqabi wnf nfcr
+ sbcbii wb alimi sbcnestgf sylan2 bitrid abbidv sylan 3eqtr4g ) CFHZAEIZBJ
+ ZKGLZBDEMZHZACNZGOZUOEHZBACDMZNZGOZACUPMBVAEMULCPHZUNUSVCQCFRVDUNKZURVBGU
+ RUTBDNZACNZVEVBUQVFACVFGUPBGDESTUCUNVDUTAUAZBJVGVBUDUMVHBAGEUBUEUTABCDPUF
+ UGUHUIUJAGCUPSBGVAESUK $.
$d x ph $.
$( Nest the composition of two substitutions. Usage of this theorem is
@@ -41570,7 +41571,7 @@ among classes ( ~ eq0 , as opposed to the weaker uniqueness among sets,
17-Feb-2004.) (Proof modification is discouraged.)
(New usage is discouraged.) $)
disjOLD $p |- ( ( A i^i B ) = (/) <-> A. x e. A -. x e. B ) $=
- ( cin c0 wceq cv wcel wn wi wal wral wa cab df-in eqeq1i abeq1 imnan noel
+ ( cin c0 wceq cv wcel wn wi wal wral wa cab df-in eqeq1i eqabc imnan noel
wb nbn bitr2i albii 3bitri df-ral bitr4i ) BCDZEFZAGZBHZUICHZIZJZAKZULABL
UHUJUKMZANZEFUOUIEHZTZAKUNUGUPEABCOPUOAEQURUMAUMUOIURUJUKRUQUOUISUAUBUCUD
ULABUEUF $.
@@ -49783,10 +49784,10 @@ under a function is also a set (see the variant ~ funimaex ). Although
$( A version of Replacement using class abstractions. (Contributed by NM,
26-Nov-1995.) $)
zfrep4 $p |- { y | E. x ( ph /\ ps ) } e. _V $=
- ( cv cab wcel wa wex cvv abid anbi1i exbii abbii wceq wb wal nfab1 sylbi
- wi zfrepclf abeq2 mpbir issetri eqeltrri ) CHACIZJZBKZCLZDIZABKZCLZDIMULU
- ODUKUNCUJABACNZOPQEUMEHZUMRZELDHZUQJULSDTZELBCDEUIACUAFUJABUSUQRUCDTELUPG
- UBUDURUTEULDUQUEPUFUGUH $.
+ ( cv cab wcel wa wex cvv abid anbi1i exbii abbii wceq wb wal eqab issetri
+ nfab1 wi sylbi zfrepclf mpbir eqeltrri ) CHACIZJZBKZCLZDIZABKZCLZDIMULUOD
+ UKUNCUJABACNZOPQEUMEHZUMRZELDHZUQJULSDTZELBCDEUIACUCFUJABUSUQRUDDTELUPGUE
+ UFURUTEULDUQUAPUGUBUH $.
$}
@@ -50667,8 +50668,8 @@ That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with
~ vpwex . (Revised by BJ, 10-Aug-2022.) $)
vpwex $p |- ~P x e. _V $=
( vw vy vz cv cpw wss cab cvv df-pw wceq wex wel wal axpow2 bm1.3ii sseq1
- wb abeq2w exbii mpbir issetri eqeltri ) AEZFBEZUDGZBHZIBUDJCUGCEZUGKZCLDC
- MDEZUDGZRDNZCLUKCDACDOPUIULCUFUKBDUHUEUJUDQSTUAUBUC $.
+ wb eqabw exbii mpbir issetri eqeltri ) AEZFBEZUDGZBHZIBUDJCUGCEZUGKZCLDCM
+ DEZUDGZRDNZCLUKCDACDOPUIULCUFUKBDUHUEUJUDQSTUAUBUC $.
$( $j usage 'vpwex' avoids 'ax-10' 'ax-11' 'ax-12'; $)
$}
@@ -55096,13 +55097,13 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM,
ssrelOLD $p |- ( Rel A -> ( A C_ B <->
A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) ) $=
( vz wrel wss cv cop wcel wi wal ssel alrimivv wceq wex cvv dfss2 eleq1
- wa cxp df-rel sylbb copab cab df-xp df-opab eqtri abeq2i simpl sylbi sylg
- 2eximi imim2i imbi12d biimprcd 2alimi 19.23vv sylib com23 a2d alimdv syl5
- syl6ibr com12 impbid2 ) CFZCDGZAHZBHZIZCJZVKDJZKZBLALZVHVNABCDVKMNVOVGVHV
- OVGEHZCJZVPDJZKZELZVHVGVQVPVKOZBPAPZKZELVOVTVGVQVPQQUAZJZKZWCEVGCWDGWFELC
- UBECWDRUCWEWBVQWEWAVIQJVJQJTZTZBPAPZWBWIEWDWDWGABUDWIEUEABQQUFWGABEUGUHUI
- WHWAABWAWGUJUMUKUNULVOWCVSEVOVQWBVRVOWBVQVRVOWAVSKZBLALWBVSKVNWJABWAVSVNW
- AVQVLVRVMVPVKCSVPVKDSUOUPUQWAVSABURUSUTVAVBVCECDRVDVEVF $.
+ wa cxp df-rel sylbb copab cab df-xp eqtri eqabi simpl 2eximi sylbi imim2i
+ df-opab sylg imbi12d biimprcd 2alimi 19.23vv sylib com23 a2d syl5 syl6ibr
+ alimdv com12 impbid2 ) CFZCDGZAHZBHZIZCJZVKDJZKZBLALZVHVNABCDVKMNVOVGVHVO
+ VGEHZCJZVPDJZKZELZVHVGVQVPVKOZBPAPZKZELVOVTVGVQVPQQUAZJZKZWCEVGCWDGWFELCU
+ BECWDRUCWEWBVQWEWAVIQJVJQJTZTZBPAPZWBWIEWDWDWGABUDWIEUEABQQUFWGABEUMUGUHW
+ HWAABWAWGUIUJUKULUNVOWCVSEVOVQWBVRVOWBVQVRVOWAVSKZBLALWBVSKVNWJABWAVSVNWA
+ VQVLVRVMVPVKCSVPVKDSUOUPUQWAVSABURUSUTVAVDVBECDRVCVEVF $.
$( Extensionality principle for relations. Theorem 3.2(ii) of [Monk1]
p. 33. (Contributed by NM, 2-Aug-1994.) $)
@@ -55460,14 +55461,14 @@ of the restricted converse (see ~ cnvrescnv ). (Contributed by NM,
(Revised by KP, 25-Oct-2021.) $)
relopabi $p |- Rel A $=
( vz vu cvv cv wcel wex cop wceq wa copab cab df-opab eqtri 2eximi syl
- wrel cxp abeq2i simpl sylbi ax6evr pm3.21 eximdv opeq2 eqtr2 eqcomd sylan
- wss mpi eximi eqcoms excomim vex pm3.2i jctr df-xp sylibr 3syl ax5e ssriv
- df-rel mpbir ) DUADHHUBZUMFDVHFIZDJZVIVHJZCKZVKVJVIBIZCIZLZMZCKBKZVIVMGIZ
- LZMZGKZBKZCKZVLVJVPANZCKBKZVQWEFDDABCOWEFPEABCFQRUCWDVPBCVPAUDSUEVQWACKBK
- WCVPWABCWAVOVIVOVIMZVNVRMZWFNZGKZWAWFWGGKWIGCUFWFWGWHGWFWGUGUHUNWHVTGWGVO
- VSMZWFVTVNVRVMUIWJWFNVSVIVOVSVIUJUKULUOTUPSWABCUQTWBVKCWBVTVMHJZVRHJZNZNZ
- GKBKZVKVTWNBGVTWMWKWLBURGURUSUTSWOFVHVHWMBGOWOFPBGHHVAWMBGFQRUCVBUOVCVKCV
- DTVEDVFVG $.
+ wrel cxp wss eqabi simpl sylbi ax6evr pm3.21 eximdv mpi opeq2 eqtr2 sylan
+ eqcomd eximi eqcoms excomim vex pm3.2i jctr df-xp sylibr 3syl ax5e df-rel
+ ssriv mpbir ) DUADHHUBZUCFDVHFIZDJZVIVHJZCKZVKVJVIBIZCIZLZMZCKBKZVIVMGIZL
+ ZMZGKZBKZCKZVLVJVPANZCKBKZVQWEFDDABCOWEFPEABCFQRUDWDVPBCVPAUESUFVQWACKBKW
+ CVPWABCWAVOVIVOVIMZVNVRMZWFNZGKZWAWFWGGKWIGCUGWFWGWHGWFWGUHUIUJWHVTGWGVOV
+ SMZWFVTVNVRVMUKWJWFNVSVIVOVSVIULUNUMUOTUPSWABCUQTWBVKCWBVTVMHJZVRHJZNZNZG
+ KBKZVKVTWNBGVTWMWKWLBURGURUSUTSWOFVHVHWMBGOWOFPBGHHVAWMBGFQRUDVBUOVCVKCVD
+ TVFDVEVG $.
$( $j usage 'relopabi' avoids 'ax-sep' 'ax-nul' 'ax-pr'; $)
$( Alternate proof of ~ relopabi (shorter but uses more axioms).
@@ -56400,9 +56401,9 @@ ordered pairs as classes (in set.mm, the Kuratowski encoding). A more
dmopab3 $p |- ( A. x e. A E. y ph <->
dom { <. x , y >. | ( x e. A /\ ph ) } = A ) $=
( wex wral cv wcel wi wal wa wb copab cdm wceq df-ral pm4.71 albii dmopab
- cab 19.42v abbii eqtri eqeq1i eqcom abeq2 3bitr2ri 3bitri ) ACEZBDFBGDHZU
- IIZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCE
- ZBTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
+ cab 19.42v abbii eqtri eqeq1i eqcom eqab 3bitr2ri 3bitri ) ACEZBDFBGDHZUI
+ IZBJUJUJUIKZLZBJZUJAKZBCMNZDOZUIBDPUKUMBUJUIQRUQULBTZDODUROUNUPURDUPUOCEZ
+ BTURUOBCSUSULBUJACUAUBUCUDDURUEULBDUFUGUH $.
$}
${
@@ -56444,7 +56445,7 @@ ordered pairs as classes (in set.mm, the Kuratowski encoding). A more
$( An empty domain is equivalent to an empty range. (Contributed by NM,
21-May-1998.) $)
dm0rn0 $p |- ( dom A = (/) <-> ran A = (/) ) $=
- ( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii abeq1 3bitr4i
+ ( vx vy cv wbr wex cab c0 wceq wcel wb wal alnex noel albii eqabc 3bitr4i
wn nbn eqeq1i cdm crn excom xchbinx bitr4i 3bitr3i df-dm dfrn2 ) BDZCDZAE
ZCFZBGZHIZUKBFZCGZHIZAUAZHIAUBZHIULUIHJZKZBLZUOUJHJZKZCLZUNUQULRZBLZUORZC
LZVBVEVGUOCFZRVIVGULBFVJULBMUKBCUCUDUOCMUEVFVABUTULUINSOVHVDCVCUOUJNSOUFU
@@ -58254,8 +58255,8 @@ the restriction (of the relation) to the singleton containing this
25-Oct-2021.) $)
cnv0 $p |- `' (/) = (/) $=
( vx vz vy c0 ccnv cv wcel cop wbr wa wex br0 intnan nex copab cab df-cnv
- wceq df-opab eqtri abeq2i mtbir nel0 ) ADEZAFZUDGUEBFZCFZHRZUGUFDIZJZCKZB
- KZUKBUJCUIUHUGUFLMNNULAUDUDUIBCOULAPBCDQUIBCASTUAUBUC $.
+ wceq df-opab eqtri eqabi mtbir nel0 ) ADEZAFZUDGUEBFZCFZHRZUGUFDIZJZCKZBK
+ ZUKBUJCUIUHUGUFLMNNULAUDUDUIBCOULAPBCDQUIBCASTUAUBUC $.
$( The converse of the identity relation. Theorem 3.7(ii) of [Monk1]
p. 36. (Contributed by NM, 26-Apr-1998.) (Proof shortened by Andrew
@@ -62710,11 +62711,11 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
$( A condition showing a class is single-rooted. (See ~ funcnv ).
(Contributed by NM, 26-May-2006.) $)
funcnv3 $p |- ( Fun `' A <-> A. y e. ran A E! x e. dom A x A y ) $=
- ( cv wbr wmo crn wral wex ccnv wfun cdm wreu wcel dfrn2 abeq2i biimpi weu
- wa vex biantrurd ralbiia funcnv df-reu breldm pm4.71ri eubii df-eu ralbii
- 3bitr2i 3bitr4i ) ADZBDZCEZAFZBCGZHUNAIZUOSZBUPHCJKUNACLZMZBUPHUOURBUPUMU
- PNZUQUOVAUQUQBUPABCOPQUAUBABCUCUTURBUPUTULUSNZUNSZARUNARURUNAUSUDUNVCAUNV
- BULUMCATBTUEUFUGUNAUHUJUIUK $.
+ ( cv wbr wmo crn wral wex wa ccnv wfun cdm wreu wcel dfrn2 biimpi weu vex
+ eqabi biantrurd ralbiia funcnv df-reu breldm pm4.71ri eubii df-eu 3bitr2i
+ ralbii 3bitr4i ) ADZBDZCEZAFZBCGZHUNAIZUOJZBUPHCKLUNACMZNZBUPHUOURBUPUMUP
+ OZUQUOVAUQUQBUPABCPTQUAUBABCUCUTURBUPUTULUSOZUNJZARUNARURUNAUSUDUNVCAUNVB
+ ULUMCASBSUEUFUGUNAUHUIUJUK $.
$( The double converse of a class is a function iff the class is
single-valued. Each side is equivalent to Definition 6.4(2) of
@@ -62904,11 +62905,11 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
(New usage is discouraged.) $)
funimaexgOLD $p |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V ) $=
( vw vx vy vz wcel wfun cima cvv cv wceq imaeq2 eleq1d wal wex wel bitri
- wi imbi2d cop dffun5 wa wb nfv axrep4 isset cab dfima3 eqeq2i abeq2 exbii
- wrel sylibr simplbiim vtoclg impcom ) BCHAIZABJZKHZUSADLZJZKHZTUSVATDBCVB
- BMZVDVAUSVEVCUTKVBBANOUAUSAUNELFLZUBAHZVFGLZMTFPGQEPZVDEFGAUCVIFGREDRVGUD
- EQZUEFPZGQZVDVGEFGDVGGUFUGVDVHVCMZGQVLGVCUHVMVKGVMVHVJFUIZMVKVCVNVHEFAVBU
- JUKVJFVHULSUMSUOUPUQUR $.
+ wi imbi2d wrel cop dffun5 wa wb nfv axrep4 isset dfima3 eqeq2i eqab exbii
+ cab sylibr simplbiim vtoclg impcom ) BCHAIZABJZKHZUSADLZJZKHZTUSVATDBCVBB
+ MZVDVAUSVEVCUTKVBBANOUAUSAUBELFLZUCAHZVFGLZMTFPGQEPZVDEFGAUDVIFGREDRVGUEE
+ QZUFFPZGQZVDVGEFGDVGGUGUHVDVHVCMZGQVLGVCUIVMVKGVMVHVJFUNZMVKVCVNVHEFAVBUJ
+ UKVJFVHULSUMSUOUPUQUR $.
$}
${
@@ -63303,13 +63304,13 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
mptfnf $p |- ( A. x e. A B e. _V <-> ( x e. A |-> B ) Fn A ) $=
( vy wcel wral cv wceq wfn ralbii wex wmo wa wal wi albii df-ral 3bitr4ri
cab cvv weu cmpt eueq r19.26 df-eu copab wfun df-mpt fneq1i df-fn moanimv
- cdm bitri funopab eqcom dmopab 19.42v abbii eqtri eqeq1i wb pm4.71 abeq2f
- 3bitr4i anbi12i ancom 3bitr2i bitr4i ) CUAFZABGEHCIZEUBZABGZABCUCZBJZVJVL
- ABECUDKVKELZVKEMZNZABGVPABGZVQABGZNZVMVOVPVQABUEVLVRABVKEUFKVOAHBFZVKNZAE
- UGZUHZWDUMZBIZNZVTVSNWAVOWDBJWHBVNWDAEBCUIUJWDBUKUNVTWEVSWGWCEMZAOWBVQPZA
- OWEVTWIWJAWBVKEULQWCAEUOVQABRSWBVPNZATZBIBWLIZWGVSWLBUPWFWLBWFWCELZATWLWC
- AEUQWNWKAWBVKEURUSUTVAWBVPPZAOWBWKVBZAOVSWMWOWPAWBVPVCQVPABRWKABDVDVESVFV
- TVSVGVHSVI $.
+ cdm bitri funopab eqcom dmopab 19.42v abbii eqtri eqeq1i wb eqabf 3bitr4i
+ pm4.71 anbi12i ancom 3bitr2i bitr4i ) CUAFZABGEHCIZEUBZABGZABCUCZBJZVJVLA
+ BECUDKVKELZVKEMZNZABGVPABGZVQABGZNZVMVOVPVQABUEVLVRABVKEUFKVOAHBFZVKNZAEU
+ GZUHZWDUMZBIZNZVTVSNWAVOWDBJWHBVNWDAEBCUIUJWDBUKUNVTWEVSWGWCEMZAOWBVQPZAO
+ WEVTWIWJAWBVKEULQWCAEUOVQABRSWBVPNZATZBIBWLIZWGVSWLBUPWFWLBWFWCELZATWLWCA
+ EUQWNWKAWBVKEURUSUTVAWBVPPZAOWBWKVBZAOVSWMWOWPAWBVPVEQVPABRWKABDVCVDSVFVT
+ VSVGVHSVI $.
$( The maps-to notation defines a function with domain. (Contributed by
NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.) $)
@@ -65613,11 +65614,11 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
E. x e. B ( F ` x ) = C ) ) $=
( vy wfn wss wa cima wcel cv cfv wceq wrex cvv anim2i eleq1 wb wi rexbidv
elex fvex mpbii rexlimivw eqeq2 bibi12d imbi2d wfun cdm fnfun fndm sseq2d
- cab biimpar dfimafn syl2an2r abeq2d vtoclg impcom pm5.21nd ) EBGZCBHZIZDE
- CJZKZALZEMZDNZACOZVDDPKZIVFVKVDDVEUBQVJVKVDVIVKACVIVHPKVKVGEUCVHDPRUDUEQV
- KVDVFVJSZVDFLZVEKZVHVMNZACOZSZTVDVLTFDPVMDNZVQVLVDVRVNVFVPVJVMDVERVRVOVIA
- CVMDVHUFUAUGUHVDVPFVEVBEUIVCCEUJZHZVEVPFUNNBEUKVBVTVCVBVSBCBEULUMUOAFCEUP
- UQURUSUTVA $.
+ cab biimpar dfimafn syl2an2r eqabd vtoclg impcom pm5.21nd ) EBGZCBHZIZDEC
+ JZKZALZEMZDNZACOZVDDPKZIVFVKVDDVEUBQVJVKVDVIVKACVIVHPKVKVGEUCVHDPRUDUEQVK
+ VDVFVJSZVDFLZVEKZVHVMNZACOZSZTVDVLTFDPVMDNZVQVLVDVRVNVFVPVJVMDVERVRVOVIAC
+ VMDVHUFUAUGUHVDVPFVEVBEUIVCCEUJZHZVEVPFUNNBEUKVBVTVCVBVSBCBEULUMUOAFCEUPU
+ QURUSUTVA $.
$}
${
@@ -65707,11 +65708,11 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
( cv cfv cio wceq cdm fveq2 iotabidv eqeq12d wcel vex sylib sylbi wbr wex
eldm nfiota1 nfeq2 wfun opabiotafun funbrfv ax-mp cab csn cop copab df-br
wi eleq2i opabidw 3bitri vsnid id eleqtrrid abid weu wb breldm opabiotadm
- abeq2i iota1 syl mpbid eqtr4d exlimi vtoclga ) CIZFJZADKZLZEFJZBDKZLCEFMZ
- VNELZVOVRVPVSVNEFNWAABDHOPVNVTQZVNDIZFUAZDUBVQDVNFCRZUCWDVQDDVOVPADUDUEWD
- VOWCVPFUFWDVOWCLUOACDFGUGVNWCFUHUIWDAVPWCLZWDADUJZWCUKZLZAWDVNWCULZFQWJWI
- CDUMZQWIVNWCFUNFWKWJGUPWICDUQURWIWCWGQAWIWCWHWGDUSWIUTVAADVBSTWDADVCZAWFV
- DWDWBWLVNWCFWEDRVEWLCVTACDFGVFVGSADVHVIVJVKVLTVM $.
+ eqabi iota1 syl mpbid eqtr4d exlimi vtoclga ) CIZFJZADKZLZEFJZBDKZLCEFMZV
+ NELZVOVRVPVSVNEFNWAABDHOPVNVTQZVNDIZFUAZDUBVQDVNFCRZUCWDVQDDVOVPADUDUEWDV
+ OWCVPFUFWDVOWCLUOACDFGUGVNWCFUHUIWDAVPWCLZWDADUJZWCUKZLZAWDVNWCULZFQWJWIC
+ DUMZQWIVNWCFUNFWKWJGUPWICDUQURWIWCWGQAWIWCWHWGDUSWIUTVAADVBSTWDADVCZAWFVD
+ WDWBWLVNWCFWEDRVEWLCVTACDFGVFVGSADVHVIVJVKVLTVM $.
$}
$( The image of a pair under a function. (Contributed by Jeff Madsen,
@@ -66145,14 +66146,14 @@ empty set when it is not meaningful (as shown by ~ ndmfv and ~ fvprc ).
` A ` . (Contributed by Mario Carneiro, 13-Feb-2015.) $)
fvmptss $p |- ( A. x e. A B C_ C -> ( F ` D ) C_ C ) $=
( vy wss wcel cfv cv wi wceq fveq2 sseq1d imbi2d nfcv wa c0 dmmptss sseli
- wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv dmmpt rabeq2i fvmpt2
- eqimss syl sylbi wn ndmfv 0ss eqsstrdi pm2.61i rsp impcom sstrid vtoclgaf
- ex vtoclga sylan2 adantl pm2.61dan ) CDIZABUCZEFUDZJZEFKZDIZVPVNEBJZVRVOB
- EABCFGUAUBVSVNVRVNHLZFKZDIZMZVNVRMHEBVTENZWBVRVNWDWAVQDVTEFOPQVNALZFKZDIZ
- MWCAVTBAVTRZVNWBAVMABUEAWADAVTFAFABCUFGABCUGUHWHUIADRUJUKWEVTNZWGWBVNWIWF
- WADWEVTFOPQWEBJZVNWGWJVNSWFCDWEVOJZWFCIZWKWJCULJZSZWLWMAVOBABCFGUMUNWNWFC
- NWLABCULFGUOWFCUPUQURWKUSWFTCWEFUTCVAVBVCVNWJVMVMABVDVEVFVHVGVIVEVJVNVPUS
- ZSVQTDWOVQTNVNEFUTVKDVAVBVL $.
+ wral cdm nfra1 cmpt nfmpt1 nfcxfr nffv nfss nfim cvv reqabi fvmpt2 eqimss
+ dmmpt syl sylbi wn ndmfv 0ss eqsstrdi pm2.61i rsp impcom vtoclgaf vtoclga
+ sstrid ex sylan2 adantl pm2.61dan ) CDIZABUCZEFUDZJZEFKZDIZVPVNEBJZVRVOBE
+ ABCFGUAUBVSVNVRVNHLZFKZDIZMZVNVRMHEBVTENZWBVRVNWDWAVQDVTEFOPQVNALZFKZDIZM
+ WCAVTBAVTRZVNWBAVMABUEAWADAVTFAFABCUFGABCUGUHWHUIADRUJUKWEVTNZWGWBVNWIWFW
+ ADWEVTFOPQWEBJZVNWGWJVNSWFCDWEVOJZWFCIZWKWJCULJZSZWLWMAVOBABCFGUPUMWNWFCN
+ WLABCULFGUNWFCUOUQURWKUSWFTCWEFUTCVAVBVCVNWJVMVMABVDVEVHVIVFVGVEVJVNVPUSZ
+ SVQTDWOVQTNVNEFUTVKDVAVBVL $.
$}
${
@@ -67229,7 +67230,7 @@ in the range of the function (the implication "to the right" is always
A. y e. B E. x e. A y = ( F ` x ) ) ) $=
( wfo wf crn wceq wa cv cfv wrex wral dffo2 cab wb wcel wal wi wfn fnrnfv
ffn eqeq1d syl dfbi2 ffvelcdm adantr eqeltrd rexlimdva2 biantrurd bitr4id
- simpr albidv abeq1 df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZDIZJV
+ simpr albidv eqabc df-ral 3bitr4g bitrd pm5.32i bitri ) CDEFCDEGZEHZDIZJV
ABKZAKZELZIZACMZBDNZJCDEOVAVCVIVAVCVHBPZDIZVIVAECUAZVCVKQCDEUCVLVBVJDABCE
UBUDUEVAVHVDDRZQZBSVMVHTZBSVKVIVAVNVOBVAVNVHVMTZVOJVOVHVMUFVAVPVOVAVGVMAC
VAVECRJZVGJVDVFDVQVGUMVQVFDRVGCDVEEUGUHUIUJUKULUNVHBDUOVHBDUPUQURUSUT $.
@@ -76276,12 +76277,12 @@ numbers have the property (conclusion). Exercise 25 of [Enderton]
tfis $p |- ( x e. On -> ph ) $=
( vz vw cv con0 wcel crab wss wi wral wceq ssrab2 wa nfcv bitr3id simprbi
wsb nfrab1 nfss nfcri nfim dfss3 sseq1 rabid eleq1w imbi12d sbequ sbequ12
- nfv nfs1v cbvrabw elrab2 ralimi syl5 anc2li vtoclgaf mp2an eqcomi rabeq2i
- rgen tfi ) BGZHIZVFAABHHABHJZHVGHKEGZVGKZVHVGIZLZEHMVGHNABHOVKEHCGZVGIZCV
- EMZVFAPZLVKBVHHBVHQZVIVJBBVHVGVPABHUAZUBBEVGVQUCUDVEVHNZVNVIVOVJVNVEVGKVR
- VICVEVGUEVEVHVGUFRVOVEVGIVRVJABHUGBEVGUHRUIVFVNAVNABCTZCVEMVFAVMVSCVEVMVL
- HIVSABFTZVSFVLHVGAFCBUJAVTBFHBHQFHQAFULABFUMABFUKUNUOSUPDUQURUSVCEVGVDUTV
- AVBS $.
+ nfs1v cbvrabw elrab2 ralimi syl5 anc2li vtoclgaf rgen mp2an eqcomi reqabi
+ nfv tfi ) BGZHIZVFAABHHABHJZHVGHKEGZVGKZVHVGIZLZEHMVGHNABHOVKEHCGZVGIZCVE
+ MZVFAPZLVKBVHHBVHQZVIVJBBVHVGVPABHUAZUBBEVGVQUCUDVEVHNZVNVIVOVJVNVEVGKVRV
+ ICVEVGUEVEVHVGUFRVOVEVGIVRVJABHUGBEVGUHRUIVFVNAVNABCTZCVEMVFAVMVSCVEVMVLH
+ IVSABFTZVSFVLHVGAFCBUJAVTBFHBHQFHQAFVCABFULABFUKUMUNSUODUPUQURUSEVGVDUTVA
+ VBS $.
$}
${
@@ -77508,15 +77509,15 @@ as a separate axiom in an axiom system (such as Kunen's) that uses this
( cv wral wrex wcel wa cvv wex cab wceq sylibr vex wi wal wmo 3imtr4i weu
copab crn wfun crab 19.42v abbii dmopab df-rab 3eqtr4i euex ralimi rabid2
cdm eqtr4id eqeltrdi eumo imim2i moanimv alimi funopab funrnex sylc nfra1
- df-ral eleq2d cop opabidw opelrn sylbir impac eximi abeq2i df-rex syl6bir
- ex ralrimi nfopab1 nfrn nfeq2 nfcv nfopab2 rexeqf ralbid spcedv ) ACUAZBD
- FZGZACEFZHZBWGGACBFZWGIZAJZBCUBZUCZHZBWGGEKWOWHWNUNZKIWNUDZWOKIWHWQWGKWHW
- QACLZBWGUEZWGWMCLZBMWLWSJZBMWQWTXAXBBWLACUFUGWMBCUHZWSBWGUIUJWHWSBWGGWGWT
- NWFWSBWGACUKULWSBWGUMOUOZDPUPWLWFQZBRWMCSZBRWHWRXEXFBXEWLACSZQXFWFXGWLACU
- QURWLACUSOUTWFBWGVEWMBCVATKWNVBVCWHWPBWGWFBWGVDWHWLWKWQIZWPWHWQWGWKXDVFXA
- CFZWOIZAJZCLXHWPWMXKCWLAXJWLAXJWMWKXIVGWNIXJWMBCVHWKXIWNBPCPVIVJVPVKVLXAB
- WQXCVMACWOVNTVOVQWIWONWJWPBWGBWIWOBWNWMBCVRVSVTACWIWOCWIWACWNWMBCWBVSWCWD
- WE $.
+ df-ral eleq2d cop opabidw opelrn sylbir impac eximi eqabi syl6bir ralrimi
+ ex df-rex nfopab1 nfrn nfeq2 nfcv nfopab2 rexeqf ralbid spcedv ) ACUAZBDF
+ ZGZACEFZHZBWGGACBFZWGIZAJZBCUBZUCZHZBWGGEKWOWHWNUNZKIWNUDZWOKIWHWQWGKWHWQ
+ ACLZBWGUEZWGWMCLZBMWLWSJZBMWQWTXAXBBWLACUFUGWMBCUHZWSBWGUIUJWHWSBWGGWGWTN
+ WFWSBWGACUKULWSBWGUMOUOZDPUPWLWFQZBRWMCSZBRWHWRXEXFBXEWLACSZQXFWFXGWLACUQ
+ URWLACUSOUTWFBWGVEWMBCVATKWNVBVCWHWPBWGWFBWGVDWHWLWKWQIZWPWHWQWGWKXDVFXAC
+ FZWOIZAJZCLXHWPWMXKCWLAXJWLAXJWMWKXIVGWNIXJWMBCVHWKXIWNBPCPVIVJVPVKVLXABW
+ QXCVMACWOVQTVNVOWIWONWJWPBWGBWIWOBWNWMBCVRVSVTACWIWOCWIWACWNWMBCWBVSWCWDW
+ E $.
$}
$( If the domain of an onto function exists, so does its codomain.
@@ -82954,18 +82955,18 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
$( Lemma for well-founded recursion. An acceptable function is a function.
(Contributed by Paul Chapman, 21-Apr-2012.) $)
frrlem2 $p |- ( g e. B -> Fun g ) $=
- ( vz vw cv wcel wfn wss cpred wral wa cfv cres co wceq w3a frrlem1 abeq2i
- wex wfun fnfun 3ad2ant1 exlimiv sylbi ) GLZDMULJLZNZUMCOCEKLZPZUMOKUMQRZU
- OULSUOULUPTHUAUBKUMQZUCZJUFZULUGZUTGDABJKCDEFGHIUDUEUSVAJUNUQVAURUMULUHUI
- UJUK $.
+ ( vz vw cv wcel wfn wss cpred wral wa cfv cres co wceq w3a wex wfun eqabi
+ frrlem1 fnfun 3ad2ant1 exlimiv sylbi ) GLZDMULJLZNZUMCOCEKLZPZUMOKUMQRZUO
+ ULSUOULUPTHUAUBKUMQZUCZJUDZULUEZUTGDABJKCDEFGHIUGUFUSVAJUNUQVAURUMULUHUIU
+ JUK $.
$( Lemma for well-founded recursion. An acceptable function's domain is a
subset of ` A ` . (Contributed by Paul Chapman, 21-Apr-2012.) $)
frrlem3 $p |- ( g e. B -> dom g C_ A ) $=
- ( vz vw cv wcel wfn wss cpred wral wa cfv cres co wceq w3a wex cdm abeq2i
- frrlem1 fndm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUOJLZNZU
- PCOZCEKLZPZUPOKUPQZRZUSUOSUSUOUTTHUAUBKUPQZUCZJUDZUOUEZCOZVEGDABJKCDEFGHI
- UGUFVDVGJUQVBVGVCUQURVGVAUQVGURUQVFUPCUPUOUHUIUJUKULUMUN $.
+ ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq w3a wex frrlem1 eqabi
+ co cdm fndm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUOJLZNZUP
+ COZCEKLZPZUPOKUPQZRZUSUOSUSUOUTTHUFUAKUPQZUBZJUCZUOUGZCOZVEGDABJKCDEFGHIU
+ DUEVDVGJUQVBVGVCUQURVGVAUQVGURUQVFUPCUPUOUHUIUJUKULUMUN $.
$}
${
@@ -82985,27 +82986,27 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( a G ( ( g |` ( dom g i^i dom h ) ) |`
Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) $=
( vb vc cv wcel wa cres wceq wral wss cdm cin wfn cpred co frrlem2 funfnd
- cfv fnresin1 syl adantr w3a wex frrlem1 fndm raleqdv biimp3ar rsp exlimiv
- wi abeq2i sylbi elinel1 impel adantlr simpr fvresd resres predss exdistrv
- sseqin2 mpbi inss1 simpl2l sstrid simp2r nfan wel syl5com elinel2 anim12d
- nfra1 biimpi syl6com ralrimi syl2an wb simpl1 simpr1 ineq12 sseq1d sseq2d
- ssin fndmd raleqbidv anbi12d syl2anc mpbir2and exlimivv sylbir preddowncl
- syl2anb sylc eqtrid reseq2d oveq2d 3eqtr4d ralrimiva jca ) GNZDOZHNZDOZPZ
- XJXJUAZXLUAZUBZQZXQUCZJNZXRUHZXTXRXQEXTUDZQZIUEZRZJXQSXKXSXMXKXJXOUCXSXKX
- JABCDEFGIKUFUGXOXPXJUIUJUKXNYEJXQXNXTXQOZPZXTXJUHZXTXJCEXTUDZQZIUEZYAYDXK
- YFYHYKRZXMXKXTXOOZYLYFXKXJLNZUCZYNCTZYIYNTZJYNSZPZYLJYNSZULZLUMZYMYLUTZUU
- BGDABLJCDEFGIKUNVAZUUAUUCLUUAYLJXOSZUUCYOYSUUEYTYOYSPYLJXOYNYOXOYNRZYSYNX
- JUOUKUPUQYLJXOURUJUSVBXTXOXPVCVDVEYGXTXQXJXNYFVFZVGYGYCYJXTIYGYCXJXQYBUBZ
- QYJXJXQYBVHYGUUHYIXJYGUUHYBYIYBXQTUUHYBRXQEXTVIYBXQVKVLYGXQCTZYIXQTZJXQSZ
- PZYFYBYIRXNUULYFXKUUBXLMNZUCZUUMCTZYIUUMTZJUUMSZPZXTXLUHXTXLYIQIUERJUUMSZ
- ULZMUMZUULXMUUDUVAHDABMJCDEFHIKUNVAUUBUVAPUUAUUTPZMUMLUMUULUUAUUTLMVJUVBU
- ULLMUVBUULYNUUMUBZCTZYIUVCTZJUVCSZUVBUVCYNCYNUUMVMYPYRYOYTUUTVNVOUUAYRUUQ
- UVFUUTYOYPYRYTVPUUNUUOUUQUUSVPYRUUQPZUVEJUVCYRUUQJYQJYNWBUUPJUUMWBVQXTUVC
- OZUVGYQUUPPZUVEUVHYRYQUUQUUPUVHJLVRYRYQXTYNUUMVCYQJYNURVSUVHJMVRUUQUUPXTY
- NUUMVTUUPJUUMURVSWAUVIUVEYIYNUUMWMWCWDWEWFUVBUUFXPUUMRZUULUVDUVFPWGUVBYNX
- JYOYSYTUUTWHWNUVBUUMXLUUAUUNUURUUSWIWNUUFUVJPZUUIUVDUUKUVFUVKXQUVCCXOYNXP
- UUMWJZWKUVKUUJUVEJXQUVCUVLUVKXQUVCYIUVLWLWOWPWQWRWSWTXBUKUUGJCXQEXTXAXCXD
- XEXDXFXGXHXI $.
+ cfv fnresin1 syl adantr w3a wex wi frrlem1 eqabi raleqdv biimp3ar exlimiv
+ fndm rsp sylbi elinel1 impel adantlr simpr fvresd resres sseqin2 exdistrv
+ predss mpbi inss1 simpl2l sstrid simp2r nfra1 wel syl5com elinel2 anim12d
+ nfan ssin biimpi syl6com ralrimi syl2an simpl1 fndmd simpr1 ineq12 sseq1d
+ wb sseq2d raleqbidv anbi12d syl2anc mpbir2and exlimivv syl2anb preddowncl
+ sylbir sylc eqtrid reseq2d oveq2d 3eqtr4d ralrimiva jca ) GNZDOZHNZDOZPZX
+ JXJUAZXLUAZUBZQZXQUCZJNZXRUHZXTXRXQEXTUDZQZIUEZRZJXQSXKXSXMXKXJXOUCXSXKXJ
+ ABCDEFGIKUFUGXOXPXJUIUJUKXNYEJXQXNXTXQOZPZXTXJUHZXTXJCEXTUDZQZIUEZYAYDXKY
+ FYHYKRZXMXKXTXOOZYLYFXKXJLNZUCZYNCTZYIYNTZJYNSZPZYLJYNSZULZLUMZYMYLUNZUUB
+ GDABLJCDEFGIKUOUPZUUAUUCLUUAYLJXOSZUUCYOYSUUEYTYOYSPYLJXOYNYOXOYNRZYSYNXJ
+ UTUKUQURYLJXOVAUJUSVBXTXOXPVCVDVEYGXTXQXJXNYFVFZVGYGYCYJXTIYGYCXJXQYBUBZQ
+ YJXJXQYBVHYGUUHYIXJYGUUHYBYIYBXQTUUHYBRXQEXTVKYBXQVIVLYGXQCTZYIXQTZJXQSZP
+ ZYFYBYIRXNUULYFXKUUBXLMNZUCZUUMCTZYIUUMTZJUUMSZPZXTXLUHXTXLYIQIUERJUUMSZU
+ LZMUMZUULXMUUDUVAHDABMJCDEFHIKUOUPUUBUVAPUUAUUTPZMUMLUMUULUUAUUTLMVJUVBUU
+ LLMUVBUULYNUUMUBZCTZYIUVCTZJUVCSZUVBUVCYNCYNUUMVMYPYRYOYTUUTVNVOUUAYRUUQU
+ VFUUTYOYPYRYTVPUUNUUOUUQUUSVPYRUUQPZUVEJUVCYRUUQJYQJYNVQUUPJUUMVQWBXTUVCO
+ ZUVGYQUUPPZUVEUVHYRYQUUQUUPUVHJLVRYRYQXTYNUUMVCYQJYNVAVSUVHJMVRUUQUUPXTYN
+ UUMVTUUPJUUMVAVSWAUVIUVEYIYNUUMWCWDWEWFWGUVBUUFXPUUMRZUULUVDUVFPWMUVBYNXJ
+ YOYSYTUUTWHWIUVBUUMXLUUAUUNUURUUSWJWIUUFUVJPZUUIUVDUUKUVFUVKXQUVCCXOYNXPU
+ UMWKZWLUVKUUJUVEJXQUVCUVLUVKXQUVCYIUVLWNWOWPWQWRWSXBWTUKUUGJCXQEXTXAXCXDX
+ EXDXFXGXHXI $.
$}
${
@@ -83046,7 +83047,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and
( vw vg va cv wcel wex wss wa exlimiv cdm cop cpred vex wfn wral cfv cres
eldm2 wceq w3a cab cuni frrlem5 frrlem1 unieqi eqtri eleq2i eluniab bitri
co wel simpr2r opeldm adantr simpr1 fndmd eleqtrd rsp sylc sseqtrrd 19.8a
- abeq2i sylibr adantl elssuni syl sseqtrrdi dmss sstrd expcom impcom sylbi
+ eqabi sylibr adantl elssuni syl sseqtrrdi dmss sstrd expcom impcom sylbi
wi ) COZHUAZPWELOZUBZHPZLQDFWEUCZWFRZLWEHCUDZUIWIWKLWIWHMOZPZWMNOZUEZWODR
ZWJWORZCWOUFZSZWEWMUGWEWMWJUHIVAUJCWOUFZUKZNQZSZMQZWKWIWHXCMULZUMZPXEHXGW
HHEUMZXGABDEFGHIJKUNZEXFABNCDEFGMIJUOZUPUQURXCMWHUSUTXDWKMXCWNWKXBWNWKWDN
@@ -83406,19 +83407,19 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
( F |` Pred ( R , A , X ) ) e. _V ) $=
( vg vf vx vy vz vw wcel wa cv wss wral cres wceq cvv wfr wpo wse w3a cdm
cfv cop wfn cpred wex cab wfun fprfung funfvop sylan cuni cfrecs df-frecs
- eqtri eleq2i eluni bitri sylib eqid frrlem1 abeq2i biimpi adantl ad2antrr
- co 3simpa simprlr elssuni syl sseqtrrdi predeq3 sseq1d simprrr wbr simplr
- simprll df-br sylibr breldmg mp3an2 syl2anc simprrl fndmd eleqtrd rspcdva
- fvex sseqtrrd fun2ssres syl3anc resex eqeltrdi expr syl5 exlimdv exlimddv
+ co eqtri eleq2i eluni bitri sylib eqid frrlem1 eqabi biimpi adantl 3simpa
+ ad2antrr simprlr elssuni syl sseqtrrdi predeq3 sseq1d simprrr wbr simprll
+ simplr df-br sylibr breldmg mp3an2 syl2anc simprrl fndmd eleqtrd sseqtrrd
+ fvex rspcdva fun2ssres syl3anc resex eqeltrdi expr syl5 exlimdv exlimddv
vex mpd ) ABUAABUBABUCUDZECUEZMZNZEECUFZUGZGOZMZXIHOZIOZUHXLAPABJOZUIZXLP
- JXLQNXMXKUFXMXKXNRDVJSJXLQUDIUJHUKZMZNZCABEUIZRZTMZGXFXHCMZXQGUJZXCCULZXE
- YAABCDFUMZECUNUOYAXHXOUPZMYBCYEXHCABDUQYEFIJABHDURUSZUTGXHXOVAVBVCXFXQNZX
- IKOZUHZYHAPZABLOZUIZYHPZLYHQZNZYKXIUFYKXIYLRDVJSLYHQZUDZKUJZXTXQYRXFXPYRX
- JXPYRYRGXOIJKLAXOBHGDXOVDVEVFVGVHVHYGYQXTKYQYIYONZYGXTYIYOYPVKXFXQYSXTXFX
- QYSNZNZXSXIXRRZTUUAYCXICPXRXIUEZPXSUUBSXCYCXEYTYDVIUUAXIYECUUAXPXIYEPXFXJ
+ JXLQNXMXKUFXMXKXNRDUSSJXLQUDIUJHUKZMZNZCABEUIZRZTMZGXFXHCMZXQGUJZXCCULZXE
+ YAABCDFUMZECUNUOYAXHXOUPZMYBCYEXHCABDUQYEFIJABHDURUTZVAGXHXOVBVCVDXFXQNZX
+ IKOZUHZYHAPZABLOZUIZYHPZLYHQZNZYKXIUFYKXIYLRDUSSLYHQZUDZKUJZXTXQYRXFXPYRX
+ JXPYRYRGXOIJKLAXOBHGDXOVEVFVGVHVIVIYGYQXTKYQYIYONZYGXTYIYOYPVJXFXQYSXTXFX
+ QYSNZNZXSXIXRRZTUUAYCXICPXRXIUEZPXSUUBSXCYCXEYTYDVKUUAXIYECUUAXPXIYEPXFXJ
XPYSVLXIXOVMVNYFVOUUAXRYHUUCUUAYMXRYHPLYHEYKESYLXRYHABYKEVPVQYTYNXFXQYIYJ
- YNVRVHUUAEUUCYHUUAXEEXGXIVSZEUUCMZXCXEYTVTUUAXJUUDXFXJXPYSWAEXGXIWBWCXEXG
- TMUUDUUEECWKEXGXDTXIWDWEWFUUAYHXIXFXQYIYOWGWHZWIWJUUFWLXRCXIWMWNXIXRGXAWO
+ YNVRVIUUAEUUCYHUUAXEEXGXIVSZEUUCMZXCXEYTWAUUAXJUUDXFXJXPYSVTEXGXIWBWCXEXG
+ TMUUDUUEECWKEXGXDTXIWDWEWFUUAYHXIXFXQYIYOWGWHZWIWLUUFWJXRCXIWMWNXIXRGXAWO
WPWQWRWSXBWT $.
$( $j usage 'fprresex' avoids 'ax-rep'; $)
$}
@@ -83608,20 +83609,20 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
(Proof modification is discouraged.) (Contributed by Scott Fenton,
21-Apr-2011.) $)
wfrlem2OLD $p |- ( g e. B -> Fun g ) $=
- ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq w3a wfrlem1OLD abeq2i
- wex wfun fnfun 3ad2ant1 exlimiv sylbi ) GLZDMUKJLZNZULCOCEKLZPZULOKULQRZU
- NUKSUKUOTHSUAKULQZUBZJUEZUKUFZUSGDABJKCDEFGHIUCUDURUTJUMUPUTUQULUKUGUHUIU
- J $.
+ ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq wfun wfrlem1OLD eqabi
+ w3a wex fnfun 3ad2ant1 exlimiv sylbi ) GLZDMUKJLZNZULCOCEKLZPZULOKULQRZUN
+ UKSUKUOTHSUAKULQZUEZJUFZUKUBZUSGDABJKCDEFGHIUCUDURUTJUMUPUTUQULUKUGUHUIUJ
+ $.
$( Lemma for well-ordered recursion. An acceptable function's domain is a
subset of ` A ` . Obsolete as of 18-Nov-2024.
(New usage is discouraged.) (Proof modification is discouraged.)
(Contributed by Scott Fenton, 21-Apr-2011.) $)
wfrlem3OLD $p |- ( g e. B -> dom g C_ A ) $=
- ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq w3a wfrlem1OLD abeq2i
- wex cdm fndm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUNJLZNZU
- OCOZCEKLZPZUOOKUOQZRZURUNSUNUSTHSUAKUOQZUBZJUEZUNUFZCOZVDGDABJKCDEFGHIUCU
- DVCVFJUPVAVFVBUPUQVFUTUPVFUQUPVEUOCUOUNUGUHUIUJUKULUM $.
+ ( vz vw cv wcel wfn wss cpred wral wa cfv cres wceq wfrlem1OLD eqabi fndm
+ w3a wex cdm sseq1d biimpar adantrr 3adant3 exlimiv sylbi ) GLZDMUNJLZNZUO
+ COZCEKLZPZUOOKUOQZRZURUNSUNUSTHSUAKUOQZUEZJUFZUNUGZCOZVDGDABJKCDEFGHIUBUC
+ VCVFJUPVAVFVBUPUQVFUTUPVFUQUPVEUOCUOUNUDUHUIUJUKULUM $.
$d G f x y z w $. $d G g $.
wfrlem3OLDa.2 $e |- G e. _V $.
@@ -83657,27 +83658,27 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
( F ` ( ( g |` ( dom g i^i dom h ) ) |`
Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) $=
( vb vc wcel wa cfv wceq wral wss wex cv cdm cres cpred wfrlem2OLD funfnd
- cin wfn fnresin1 syl adantr w3a wi wfrlem1OLD abeq2i fndm raleqdv biimpar
- rsp 3adant2 exlimiv sylbi elinel1 impel adantlr adantl resres predss mpbi
- fvres sseqin2 3an6 2exbii exdistrv bitri ssinss1 ad2antrr syl5com elinel2
- nfra1 nfan anim12d biimpi syl6com ralrimi ad2ant2l jca wb ineqan12d sseq1
- ssin raleqbi1dv anbi12d imbi2d mpbiri imp 3adant3 exlimivv sylbir syl2anb
- sseq2 simpr preddowncl sylc eqtrid reseq2d fveq2d 3eqtr4d ralrimiva ) GUA
- ZDNZHUAZDNZOZXJXJUBZXLUBZUGZUCZXQUHZJUAZXRPZXRXQEXTUDZUCZIPZQZJXQRXKXSXMX
- KXJXOUHXSXKXJABCDEFGIKUEUFXOXPXJUIUJUKXNYEJXQXNXTXQNZOZXTXJPZXJCEXTUDZUCZ
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- YMYLUMZUUBGDABLJCDEFGIKUNUOZUUAUUCLYOYTUUCYSYOYTOYLJXORZUUCYOUUEYTYOYLJXO
- YNYNXJUPZUQURYLJXOUSUJUTVAVBXTXOXPVCVDVEYFYAYHQXNXTXQXJVJVFYGYCYJIYGYCXJX
- QYBUGZUCYJXJXQYBVGYGUUGYIXJYGUUGYBYIYBXQSUUGYBQXQEXTVHYBXQVKVIYGXQCSZYIXQ
- SZJXQRZOZYFYBYIQXNUUKYFXKUUBXLMUAZUHZUULCSZYIUULSZJUULRZOZXTXLPXLYIUCIPQJ
- UULRZULZMTZUUKXMUUDUUTHDABMJCDEFHIKUNUOUUBUUTOZYOUUMOZYSUUQOZYTUUROZULZMT
- LTZUUKUVFUUAUUSOZMTLTUVAUVEUVGLMYOUUMYSUUQYTUURVLVMUUAUUSLMVNVOUVEUUKLMUV
- BUVCUUKUVDUVBUVCUUKUVBUVCUUKUMZUVCYNUULUGZCSZYIUVISZJUVIRZOZUMZUVCUVJUVLY
- PUVJYRUUQYNUULCVPVQYRUUPUVLYPUUNYRUUPOZUVKJUVIYRUUPJYQJYNVTUUOJUULVTWAXTU
- VINZUVOYQUUOOZUVKUVPYRYQUUPUUOUVPXTYNNYRYQXTYNUULVCYQJYNUSVRUVPXTUULNUUPU
- UOXTYNUULVSUUOJUULUSVRWBUVQUVKYIYNUULWKWCWDWEWFWGUVBXQUVIQZUVHUVNWHYOUUMX
- OYNXPUULUUFUULXLUPWIUVRUUKUVMUVCUVRUUHUVJUUJUVLXQUVICWJUUIUVKJXQUVIXQUVIY
- IXAWLWMWNUJWOWPWQWRWSWTUKXNYFXBJCXQEXTXCXDXEXFXEXGXHXIWG $.
+ cin wfn fnresin1 syl adantr w3a wfrlem1OLD eqabi fndm raleqdv biimpar rsp
+ wi 3adant2 exlimiv sylbi elinel1 impel adantlr fvres adantl resres predss
+ sseqin2 mpbi 2exbii exdistrv bitri ssinss1 ad2antrr nfra1 syl5com elinel2
+ 3an6 nfan anim12d ssin biimpi syl6com ralrimi ad2ant2l wb ineqan12d sseq1
+ jca sseq2 raleqbi1dv anbi12d imbi2d mpbiri 3adant3 exlimivv syl2anb simpr
+ imp sylbir preddowncl sylc eqtrid reseq2d fveq2d 3eqtr4d ralrimiva ) GUAZ
+ DNZHUAZDNZOZXJXJUBZXLUBZUGZUCZXQUHZJUAZXRPZXRXQEXTUDZUCZIPZQZJXQRXKXSXMXK
+ XJXOUHXSXKXJABCDEFGIKUEUFXOXPXJUIUJUKXNYEJXQXNXTXQNZOZXTXJPZXJCEXTUDZUCZI
+ PZYAYDXKYFYHYKQZXMXKXTXONZYLYFXKXJLUAZUHZYNCSZYIYNSZJYNRZOZYLJYNRZULZLTZY
+ MYLUSZUUBGDABLJCDEFGIKUMUNZUUAUUCLYOYTUUCYSYOYTOYLJXORZUUCYOUUEYTYOYLJXOY
+ NYNXJUOZUPUQYLJXOURUJUTVAVBXTXOXPVCVDVEYFYAYHQXNXTXQXJVFVGYGYCYJIYGYCXJXQ
+ YBUGZUCYJXJXQYBVHYGUUGYIXJYGUUGYBYIYBXQSUUGYBQXQEXTVIYBXQVJVKYGXQCSZYIXQS
+ ZJXQRZOZYFYBYIQXNUUKYFXKUUBXLMUAZUHZUULCSZYIUULSZJUULRZOZXTXLPXLYIUCIPQJU
+ ULRZULZMTZUUKXMUUDUUTHDABMJCDEFHIKUMUNUUBUUTOZYOUUMOZYSUUQOZYTUUROZULZMTL
+ TZUUKUVFUUAUUSOZMTLTUVAUVEUVGLMYOUUMYSUUQYTUURVTVLUUAUUSLMVMVNUVEUUKLMUVB
+ UVCUUKUVDUVBUVCUUKUVBUVCUUKUSZUVCYNUULUGZCSZYIUVISZJUVIRZOZUSZUVCUVJUVLYP
+ UVJYRUUQYNUULCVOVPYRUUPUVLYPUUNYRUUPOZUVKJUVIYRUUPJYQJYNVQUUOJUULVQWAXTUV
+ INZUVOYQUUOOZUVKUVPYRYQUUPUUOUVPXTYNNYRYQXTYNUULVCYQJYNURVRUVPXTUULNUUPUU
+ OXTYNUULVSUUOJUULURVRWBUVQUVKYIYNUULWCWDWEWFWGWKUVBXQUVIQZUVHUVNWHYOUUMXO
+ YNXPUULUUFUULXLUOWIUVRUUKUVMUVCUVRUUHUVJUUJUVLXQUVICWJUUIUVKJXQUVIXQUVIYI
+ WLWMWNWOUJWPXAWQWRXBWSUKXNYFWTJCXQEXTXCXDXEXFXEXGXHXIWK $.
$}
${
@@ -83755,15 +83756,15 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
wfrdmclOLD $p |- ( X e. dom F -> Pred ( R , A , X ) C_ dom F ) $=
( vg vf vx vy vz vw cdm wcel cpred cv wss wral wa cfv cres wceq ciun wrex
wfn w3a wex cab cwrecs dfwrecsOLD eqtri dmeqi dmuni eleq2i eliun bitri wi
- cuni eqid wfrlem1OLD abeq2i predeq3 sseq1d rspccv adantl wb eleq2d sseq2d
- fndm imbi12d adantr adantrl 3adant3 exlimiv sylbi reximia ssiun sseqtrrdi
- mpbird syl ) ECMZNZABEOZGHPZIPZUEWEAQABJPZOZWEQJWERSWFWDTWDWGUADTUBJWERUF
- IUGHUHZGPZMZUCZWAWBEWJNZGWHUDZWCWKQZWBEWKNWMWAWKEWAWHURZMWKCWOCABDUIWOFIJ
- ABHDUJUKULGWHUMUKZUNGEWHWJUOUPWMWCWJQZGWHUDWNWLWQGWHWIWHNWIKPZUEZWRAQZABL
- PZOZWRQZLWRRZSZXAWITWIXBUADTUBLWRRZUFZKUGZWLWQUQZXHGWHIJKLAWHBHGDWHUSUTVA
- XGXIKWSXEXIXFWSXDXIWTWSXDSXIEWRNZWCWRQZUQZXDXLWSXCXKLEWRXAEUBXBWCWRABXAEV
- BVCVDVEWSXIXLVFXDWSWLXJWQXKWSWJWREWRWIVIZVGWSWJWRWCXMVHVJVKVSVLVMVNVOVPGW
- HWJWCVQVTVOWPVR $.
+ cuni eqid wfrlem1OLD eqabi predeq3 sseq1d rspccv adantl wb eleq2d imbi12d
+ fndm sseq2d adantr mpbird adantrl 3adant3 exlimiv sylbi reximia ssiun syl
+ sseqtrrdi ) ECMZNZABEOZGHPZIPZUEWEAQABJPZOZWEQJWERSWFWDTWDWGUADTUBJWERUFI
+ UGHUHZGPZMZUCZWAWBEWJNZGWHUDZWCWKQZWBEWKNWMWAWKEWAWHURZMWKCWOCABDUIWOFIJA
+ BHDUJUKULGWHUMUKZUNGEWHWJUOUPWMWCWJQZGWHUDWNWLWQGWHWIWHNWIKPZUEZWRAQZABLP
+ ZOZWRQZLWRRZSZXAWITWIXBUADTUBLWRRZUFZKUGZWLWQUQZXHGWHIJKLAWHBHGDWHUSUTVAX
+ GXIKWSXEXIXFWSXDXIWTWSXDSXIEWRNZWCWRQZUQZXDXLWSXCXKLEWRXAEUBXBWCWRABXAEVB
+ VCVDVEWSXIXLVFXDWSWLXJWQXKWSWJWREWRWIVIZVGWSWJWRWCXMVJVHVKVLVMVNVOVPVQGWH
+ WJWCVRVSVPWPVT $.
$}
${
@@ -84755,9 +84756,9 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
functions, and ` F ` is their union. First we show that an acceptable
function is in fact a function. (Contributed by NM, 9-Apr-1995.) $)
tfrlem4 $p |- ( g e. A -> Fun g ) $=
- ( vz vw cv wcel wfn cfv cres wceq wral wa con0 wrex wfun abeq2i rexlimivw
- tfrlem3 fnfun adantr sylbi ) EJZCKUGHJZLZIJZUGMUGUJNFMOIUHPZQZHRSZUGTZUME
- CABHICDEFGUCUAULUNHRUIUNUKUHUGUDUEUBUF $.
+ ( vz vw cv wcel wfn cfv cres wceq wral wa con0 wrex wfun adantr rexlimivw
+ tfrlem3 eqabi fnfun sylbi ) EJZCKUGHJZLZIJZUGMUGUJNFMOIUHPZQZHRSZUGTZUMEC
+ ABHICDEFGUCUDULUNHRUIUNUKUHUGUEUAUBUF $.
$( Lemma for transfinite recursion. The values of two acceptable functions
are the same within their domains. (Contributed by NM, 9-Apr-1995.)
@@ -84812,12 +84813,12 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
11-Mar-2008.) $)
tfrlem8 $p |- Ord dom recs ( F ) $=
( vz vg vw cdm word cv wceq wrex cuni con0 wcel cfv rexlimiv ax-mp cab wi
- crecs wss wfn cres wral tfrlem3 abeq2i fndm adantr eleq1d biimprcd eleq1a
- wa sylbi syl abssi ssorduni wb ciun recsfval dmeqi vex dmex dfiun2 3eqtri
- dmuni ordeq mpbir ) EUCZJZKZGLZHLZJZMZHCNZGUAZOZKZVSPUDWAVRGPVQVNPQZHCVOC
- QZVPPQZVQWBUBWCVOVNUEZILZVORVOWFUFERMIVNUGZUOZGPNZWDWIHCABGICDHEFUHUIWHWD
- GPWHWDWBWHVPVNPWEVPVNMWGVNVOUJUKULUMSUPVPPVNUNUQSURVSUSTVLVTMVMWAUTVLCOZJ
- HCVPVAVTVKWJABCDEFVBVCHCVHHGCVPVOHVDVEVFVGVLVTVITVJ $.
+ crecs wss wfn cres wral tfrlem3 eqabi adantr eleq1d biimprcd sylbi eleq1a
+ wa fndm syl abssi ssorduni wb ciun recsfval dmeqi dmuni vex dfiun2 3eqtri
+ dmex ordeq mpbir ) EUCZJZKZGLZHLZJZMZHCNZGUAZOZKZVSPUDWAVRGPVQVNPQZHCVOCQ
+ ZVPPQZVQWBUBWCVOVNUEZILZVORVOWFUFERMIVNUGZUOZGPNZWDWIHCABGICDHEFUHUIWHWDG
+ PWHWDWBWHVPVNPWEVPVNMWGVNVOUPUJUKULSUMVPPVNUNUQSURVSUSTVLVTMVMWAUTVLCOZJH
+ CVPVAVTVKWJABCDEFVBVCHCVDHGCVPVOHVEVHVFVGVLVTVITVJ $.
$( Lemma for transfinite recursion. Here we compute the value of ` recs `
(the union of all acceptable functions). (Contributed by NM,
@@ -84825,21 +84826,21 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $=
tfrlem9 $p |- ( B e. dom recs ( F ) ->
( recs ( F ) ` B ) = ( F ` ( recs ( F ) |` B ) ) ) $=
( vz cdm wcel cv wex cfv cres wceq wa con0 wi syl com3l crecs cop ibi wfn
- eldm2g wral wrex cab cuni dfrecs3 eleq2i eluniab bitri wss abeq2i elssuni
- fnop rspe recsfval sseqtrrdi sylbir fveq2 reseq2 fveq2d rspcv fndm eleq2d
- eqeq12d wfun tfrlem7 funssfv mp3an1 adantrl eleq1d onelss imp31 fun2ssres
- syl6bir w3a sylan2 exbiri exp31 com34 com24 sylbird syld imp4d mpdi exp4d
- ex com4r pm2.43i imp4a rexlimdv imp exlimiv sylbi ) DFUAZIZJZDHKZUBZWRJZH
- LZDWRMZWRDNZFMZOZWTXDHDWRWSUEUCXCXHHXCXBEKZJZXIAKZUDZBKZXIMZXIXMNZFMZOZBX
- KUFZPZAQUGZPZELZXHXCXBXTEUHUIZJYBWRYCXBABEFUJUKXTEXBULUMYAXHEXJXTXHXJXSXH
- AQXJXKQJZXLXRXHXLXJYDXRXHRZXLXJYDYERRXLXJYDXLYEXLXJYDXLYERRXLXJPZYDXLXRXH
- YFDXKJZYDXSPZXHRXKDXAXIUQYGYHXIWRUNZXHYHXTYIXSAQURXTXICJZYIXTECGUOYJXICUI
- WRXICUPABCEFGUSUTVASYGYDXLXRYIXHRZYGXRXLYDYKYGXRDXIMZXIDNZFMZOZXLYDYKRZRX
- QYOBDXKXMDOZXNYLXPYNXMDXIVBYQXOYMFXMDXIVCVDVHVEXLYGYOYPXLYGDXIIZJZYOYPRXL
- YRXKDXKXIVFZVGXLYDYOYSYKXLYDYSYOYKXLYDYSYOYKRYIXLYDPZYSPZYOXHYIUUBXHYOYIU
- UBPXEYLXGYNYIYSXEYLOZUUAWRVIZYIYSUUCABCEFGVJZDWRXIVKVLVMUUBYIDYRUNZXGYNOZ
- XLYDYSUUFXLYDYRQJYSUUFRXLYRXKQYTVNYRDVOVRVPUUDYIUUFUUGUUEUUDYIUUFVSXFYMFD
- WRXIVQVDVLVTVHWATWBWCWDWETWFWDWGWHSWIWJWKWLTWMWNWOWPWQWPS $.
+ eldm2g wral wrex cab cuni dfrecs3 eleq2i eluniab bitri fnop eqabi elssuni
+ wss rspe recsfval sseqtrrdi sylbir fveq2 reseq2 fveq2d eqeq12d rspcv fndm
+ eleq2d wfun tfrlem7 funssfv mp3an1 adantrl eleq1d syl6bir imp31 fun2ssres
+ onelss w3a sylan2 exbiri exp31 com34 com24 sylbird syld imp4d exp4d com4r
+ mpdi ex pm2.43i imp4a rexlimdv imp exlimiv sylbi ) DFUAZIZJZDHKZUBZWRJZHL
+ ZDWRMZWRDNZFMZOZWTXDHDWRWSUEUCXCXHHXCXBEKZJZXIAKZUDZBKZXIMZXIXMNZFMZOZBXK
+ UFZPZAQUGZPZELZXHXCXBXTEUHUIZJYBWRYCXBABEFUJUKXTEXBULUMYAXHEXJXTXHXJXSXHA
+ QXJXKQJZXLXRXHXLXJYDXRXHRZXLXJYDYERRXLXJYDXLYEXLXJYDXLYERRXLXJPZYDXLXRXHY
+ FDXKJZYDXSPZXHRXKDXAXIUNYGYHXIWRUQZXHYHXTYIXSAQURXTXICJZYIXTECGUOYJXICUIW
+ RXICUPABCEFGUSUTVASYGYDXLXRYIXHRZYGXRXLYDYKYGXRDXIMZXIDNZFMZOZXLYDYKRZRXQ
+ YOBDXKXMDOZXNYLXPYNXMDXIVBYQXOYMFXMDXIVCVDVEVFXLYGYOYPXLYGDXIIZJZYOYPRXLY
+ RXKDXKXIVGZVHXLYDYOYSYKXLYDYSYOYKXLYDYSYOYKRYIXLYDPZYSPZYOXHYIUUBXHYOYIUU
+ BPXEYLXGYNYIYSXEYLOZUUAWRVIZYIYSUUCABCEFGVJZDWRXIVKVLVMUUBYIDYRUQZXGYNOZX
+ LYDYSUUFXLYDYRQJYSUUFRXLYRXKQYTVNYRDVRVOVPUUDYIUUFUUGUUEUUDYIUUFVSXFYMFDW
+ RXIVQVDVLVTVEWATWBWCWDWETWFWDWGWJSWHWKWIWLTWMWNWOWPWQWPS $.
$( Lemma for transfinite recursion. Without using ~ ax-rep , show that all
the restrictions of ` recs ` are sets. (Contributed by Mario Carneiro,
@@ -91967,9 +91968,9 @@ the first case of his notation (simple exponentiation) and subscript it
Axiom of Choice, see ~ ac9 . (Contributed by Mario Carneiro,
22-Jun-2016.) $)
ixpn0 $p |- ( X_ x e. A B =/= (/) -> A. x e. A B =/= (/) ) $=
- ( vf cixp c0 wne cv wcel wex wral n0 cab wfn wa df-ixp abeq2i ne0i ralimi
- cfv simplbiim exlimiv sylbi ) ABCEZFGDHZUDIZDJCFGZABKZDUDLUFUHDUFUEAHZBIA
- MNZUIUETZCIZABKZUHUJUMODUDABCDPQULUGABCUKRSUAUBUC $.
+ ( vf cixp c0 wne cv wcel wex wral n0 cab wfn cfv df-ixp eqabi ne0i ralimi
+ wa simplbiim exlimiv sylbi ) ABCEZFGDHZUDIZDJCFGZABKZDUDLUFUHDUFUEAHZBIAM
+ NZUIUEOZCIZABKZUHUJUMTDUDABCDPQULUGABCUKRSUAUBUC $.
$( The infinite Cartesian product of a family ` B ( x ) ` with an empty
member is empty. The converse of this theorem is equivalent to the
@@ -93133,15 +93134,15 @@ the first case of his notation (simple exponentiation) and subscript it
Siliprandi, 24-Dec-2020.) $)
mapsnend $p |- ( ph -> ( A ^m { B } ) ~~ A ) $=
( vz vw vy csn cmap cv cfv cvv wcel a1i wceq wa wb co ovexd wi fvexd snex
- cop 2a1i wex wrex mapsnd abeq2d anbi1d r19.41v bicomi df-rex 3bitrd fveq1
- fvsng sylancl sylan9eqr eqeq2d equcom bitrdi pm5.32da anass ancom 3bitr2d
- vex anbi2d exbidv eleq1w opeq2 sneqd anbi12d equsexvw en2d ) AHIBCKZLUAZB
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- VFQVGVJWSXATAWQXAJIWPWIWEWLWTJIBVKWPWKWCVSWPWJWBWHWACVLVMVAVNVOQUPVP $.
+ cop 2a1i wex mapsnd eqabd anbi1d r19.41v bicomi df-rex 3bitrd fveq1 fvsng
+ wrex sylancl sylan9eqr eqeq2d equcom bitrdi pm5.32da anbi2d anass 3bitr2d
+ vex ancom exbidv eleq1w opeq2 sneqd anbi12d equsexvw en2d ) AHIBCKZLUAZBC
+ HMZNZCIMZUFZKZODOOABVQLUBFVSVRPZVTOPUCAWDCVSUDQWCOPAWABPZWBUEUGAWDWAVTRZS
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$}
${
@@ -93882,11 +93883,11 @@ the Axiom of Power Sets (unlike ~ enrefg ). (Contributed by
( x C_ A /\ ( g " ( B \ ( f " x ) ) ) C_ ( A \ x ) ) } $.
$( Lemma for ~ sbth . (Contributed by NM, 22-Mar-1998.) $)
sbthlem1 $p |- U. D C_ ( A \ ( g " ( B \ ( f " U. D ) ) ) ) $=
- ( cuni cv cima cdif wss unissb wcel wa abeq2i imass2 sscon 3syl wi difss2
- ssconb exbiri syl5 pm2.43d imp sylbi elssuni sstrd mprgbir ) DIZBFJZCEJZU
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- VCMVDURMUTUSULMVAUOMVJUSDUIUSULUNRVAUOCSTUPVBUMRUQVCBSTUJUK $.
+ ( cuni cv cima cdif wss unissb wcel wa eqabi imass2 sscon 3syl wi pm2.43d
+ difss2 ssconb exbiri syl5 imp sylbi elssuni sstrd mprgbir ) DIZBFJZCEJZUL
+ KZLZKZLZMAJZURMADADURNUSDOZUSBUMCUNUSKZLZKZLZURUTUSBMZVCBUSLMZPZUSVDMZVGA
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+ CMVDURMUTUSULMVAUOMVJUSDUIUSULUNRVAUOCSTUPVBUMRUQVCBSTUJUK $.
$( Lemma for ~ sbth . (Contributed by NM, 22-Mar-1998.) $)
sbthlem2 $p |- ( ran g C_ A ->
@@ -98859,68 +98860,68 @@ intersections of elements of the argument (see ~ elfi2 ). (Contributed
sylibr pwex cxp wf adantr sstrid vex inex1 elpw rgen2 fmpo mpbi frn ssexi
nfcv cmpt mpoeq12 anidms rneqd frsucmpt sylancl sylib a1i ralrimiva sseq2
ex imbi12d sseq1 rspc2v syl3c sseld simprr onsseleq syl2an2 rzal biantrud
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- ralsn bitri syl6ibr fiin ss2ralv frnd jctild expimpd finds2 impcom simprd
- mpi a1d r19.21bi eqimss jaod sylbid ssun1 ssun2 peano2 ssiun2s 3syl eqeq1
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+ bitr3d ssfii csn id inidm eqabi ssriv simpl inss1 cbvmpov cbvmptv reseq1i
+ eqtri rdgeq1 sseqtrrid sstr2 syl5com ralimdv jctird df-suc raleqi syl6ibr
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+ simprd r19.21bi eqimss jaod sylbid ssun1 ssun2 peano2 3syl eqeq1 2rexbidv
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$}
${
@@ -99212,7 +99213,7 @@ all reals (greatest real number) for which all positive reals are greater.
dfsup2 $p |- sup ( B , A , R ) =
U. ( A \ ( ( `' R " B ) u. ( R " ( A \ ( `' R " B ) ) ) ) ) $=
( vx vy vz cv wbr wn wral wrex wi wa cdif cvv wcel vex elima bitri eqtri
- csup crab cuni ccnv cima cun df-sup cab cin dfrab3 wb abeq1 eldif mpbiran
+ csup crab cuni ccnv cima cun df-sup cab cin dfrab3 wb eqabc eldif mpbiran
wceq dfrex2 orbi12i elun ianor 3bitr4i con2bii brcnv notbii ralbii impexp
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wo invdif unieqi ) BACUADGZEGZCHZIZEBJZVTVSCHZVTFGZCHZFBKZLZEAJZMZDAUBZUC
@@ -105370,7 +105371,7 @@ of the previous layer (and the union of previous layers when the
rankf $p |- rank : U. ( R1 " On ) --> On $=
( vx vy vz cr1 con0 cima cuni crnk wf wfn cv cfv wcel wceq eqtri mpbir2an
cdm cvv mpan sylbi wral wfun csuc crab cint df-rank copab cmpt mptv dmeqi
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+ funmpt2 wex dmopab wb eqabc wrex rankwflemb intexrab isset 3bitrri mpgbir
cab df-fn c0 wne rabn0 bitr4i intex vex fvmpt2 wss ssrab2 oninton eqeltrd
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BUFZUKWBCKWENZACUGZQZVQHWHHARWEUHWHWFACWEUIOUJWIWGCULZAVBZVQWGACUMWKVQNWJ
@@ -123062,55 +123063,55 @@ divided by the argument (although we don't define full division since we
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$}
${
@@ -124581,20 +124582,20 @@ divided by the argument (although we don't define full division since we
(New usage is discouraged.) $)
1idpr $p |- ( A e. P. -> ( A .P. 1P ) = A ) $=
( vx vf vg vw vu vv vy vz wcel c1p co cv cmq wrex cltq wbr wa cnq c1q cnp
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$}
${
@@ -124735,32 +124736,32 @@ divided by the argument (although we don't define full division since we
NM, 3-Apr-1996.) (New usage is discouraged.) $)
ltexprlem1 $p |- ( B e. P. -> ( A C. B -> C =/= (/) ) ) $=
( cnp wcel wpss cv wn cplq co wa wex c0 wne pssnel prnmadd anim2i impd n0
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$( Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by
NM, 3-Apr-1996.) (New usage is discouraged.) $)
ltexprlem2 $p |- ( B e. P. -> C C. Q. ) $=
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$( Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by
NM, 6-Apr-1996.) (New usage is discouraged.) $)
ltexprlem3 $p |- ( B e. P. ->
( x e. C -> A. z ( z z e. C ) ) ) $=
( cnp wcel cv cltq wbr wi wa wn cplq co wex cnq wb elprnq cxp addnqf fdmi
- 0nnq ndmovrcl simpld 3syl prcdnq sylbid impancom anim2d eximdv abeq2i vex
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+ 3syl weq oveq2 eleq1d anbi2d exbidv elab2 3imtr4g ex com23 alrimdv ) EHIZ
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$( Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by
NM, 6-Apr-1996.) (New usage is discouraged.) $)
@@ -124770,18 +124771,18 @@ divided by the argument (although we don't define full division since we
wrex prnmax df-rex sylib brel simpld cxp addnqf fdmi ndmovrcl syl ltaddnq
ltsonq sotri sylan mpancom simprd ltexnq biimpd mpcom eqcom anim2i 19.41v
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$( Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by
NM, 6-Apr-1996.) (New usage is discouraged.) $)
@@ -124798,18 +124799,18 @@ divided by the argument (although we don't define full division since we
ltexprlem6 $p |- ( ( ( A e. P. /\ B e. P. ) /\ A C. B ) ->
( A +P. C ) C_ B ) $=
( vz vw vg vh cnp wcel co wa cv cplq wi cltq cnq vex vf wpss cpp wss wceq
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$( Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by
NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
@@ -124820,23 +124821,23 @@ B C_ ( A +P. C ) ) $=
cpp cltp wbr ltaddpr addclpr ltprord syldan mpbid pssssd sseld 2a1d com4r
wex prnmadd elprnq cxp addnqf fdmi 0nnq ndmovrcl syl simpld wrex prlem934
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$}
${
@@ -124955,78 +124956,78 @@ B C_ ( A +P. C ) ) $=
cnp wi wal wrex wral prpssnq pssnel recclnq nsmallnq adantr notbid anbi2d
wne recrecnq fvex wceq breq2 fveq2 anbi12d spcev syl6bir vex breq1 anbi1d
exbidv elab2 syl6ibr expcomd imp eximdv mpd n0 exlimiv 3syl 0pss wss prn0
- elprnq pm2.43i dmrecnq 0nnq ndmfvrcl ltrnq prcdnq biimtrid alrimiv abeq2i
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- wb ex sylbi ssriv jctil dfpss3 ltsonq sotri anim1d com12 nfe1 nfab nfcxfr
- cab nfrexw 19.8a adantll simpll expcom ltbtwnnq df-rex impcom exlimi rgen
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$( Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM,
30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
(New usage is discouraged.) $)
reclem3pr $p |- ( A e. P. -> 1P C_ ( A .P. B ) ) $=
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+ rspceeqv genpelv mpdan ad2antrr sylibrd mpd rexlimddv ex biimtrid ssrdv )
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$( Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM,
30-Apr-1996.) (New usage is discouraged.) $)
reclem4pr $p |- ( A e. P. -> ( A .P. B ) = 1P ) $=
( vw vz vf vg wcel co c1p cv c1q cltq wbr cmq wi wa cnq cnp cmp wceq wrex
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$.
$}
@@ -125947,42 +125948,42 @@ B C_ ( A +P. C ) ) $=
c0 cplr co wceq 0idsr mp1i simpl eqeltrd cop cer elexi opeq1 eceq1d df-0r
cec 1pr eqtr4di oveq2d eleq1d elab2 sylibr breq1 rspccv c1r 0lt1sr wb m1r
ne0d ltasr ax-mp mpbi m1p1sr 3brtr3i breqtri ltrelsr sotri mpan map2psrpr
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- brel simprd adantl wex sylanbr rexex wal df-ral 19.29 eleq1 bitrid notbid
- bitr3id imbi12d biimpac exlimiv syl sylanb expcom 3syl impd impancom expr
- pm2.01d ex r19.29 biimprd vex bitrdi rspcev rexbidv imbitrid imim12d a1dd
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- UMUUN $.
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+ syl6 reximdv syld rexlimivw impcom supexpr syl2anc mappsrpr sylbir simprd
+ com12 brel adantl wex sylanbr rexex wal df-ral 19.29 eleq1 bitrid bitr3id
+ notbid imbi12d biimpac exlimiv syl sylanb 3syl impd impancom pm2.01d expr
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$}
${
@@ -141733,14 +141734,14 @@ Rational numbers (as a subset of complex numbers)
( vy cv cr wcel wa cz clt co wbr c0 cle wrex syl2anc csup cdiv ssrab3 wss
cn wne wral crab cmul nnre remulcl ancoms sylan2 btwnz simpld syl cc0 zre
wb adantl simpll nngt0 jca ad2antlr ltdivmul rexbidva mpbird rabn0 sylibr
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rpnnen1lem.n $e |- NN e. _V $.
rpnnen1lem.q $e |- QQ e. _V $.
@@ -141750,42 +141751,42 @@ Rational numbers (as a subset of complex numbers)
( vy cv cr wcel cn clt co cq cz wa wbr cfv csup cdiv cmpt cmap wceq mptex
cvv fvmpt2 mpan2 wf crab ssrab2 eqsstri wss c0 wne cle wral wrex a1i cmul
remulcl ancoms sylan2 btwnz simpld syl cc0 wb zre adantl simpll nngt0 jca
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- AQNXJIHXBWBXC $.
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+ QNXJIHXBWCXC $.
$( Lemma for ~ rpnnen1 . (Contributed by Mario Carneiro, 12-May-2013.)
(Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) $)
rpnnen1lem3 $p |- ( x e. RR -> A. n e. ran ( F ` x ) n <_ x ) $=
( vy cv cr wcel cle wbr wral cn wa clt cz cfv crn csup cdiv cmpt cvv wceq
- co mptex fvmpt2 mpan2 fveq1d ovex eqid sylan9eq rabeq2i cc0 wb zre adantl
- cmul simpll nnre nngt0 jca ad2antlr ltdivmul syl3anc remulcl syl2anc ltle
- wi sylbid impr sylan2b ralrimiva wss c0 wne wrex crab eqsstri zssre sstri
- ssrab2 a1i ancoms sylan2 btwnz simpld rexbidva mpbird rabn0 sylibr neeq1i
- syl breq2 ralbidv rspcev suprleub syl31anc rpnnen1lem2 zred simpl eqbrtrd
- ledivmul cq wf wfn cmap rpnnen1lem1 elmap sylib ffn breq1 ralrn 3syl ) AK
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- BQXIYCYGURXSYBXGQXJUHMUVFABCDEFGHIXKXGQYBIHXLXMQXGYBXNYAYFDCQYBXTYEXRNXOX
- PXQWL $.
+ co mptex fvmpt2 mpan2 fveq1d ovex eqid sylan9eq cmul reqabi cc0 wb adantl
+ zre simpll nnre nngt0 ad2antlr ltdivmul syl3anc wi remulcl syl2anc sylbid
+ jca ltle impr sylan2b ralrimiva wss c0 wne wrex crab ssrab2 eqsstri zssre
+ sstri a1i ancoms sylan2 btwnz simpld syl mpbird rabn0 sylibr neeq1i breq2
+ rexbidva ralbidv rspcev suprleub syl31anc rpnnen1lem2 zred simpl ledivmul
+ eqbrtrd cq wf wfn cmap rpnnen1lem1 elmap sylib ffn breq1 ralrn 3syl ) AKZ
+ LMZDKZXRNOZDXREUAZUBPZCKZYBUAZXRNOZCQPZXSYFCQXSYDQMZRZYEBLSUCZYDUDUHZXRNX
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+ URYDSOZRZYTUUBUSYSUUCYIXTVAUTZXSYHYSVBZYHUUFXSYSYHUUDUUEYDVCZYDVDVLZVEXTX
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+ JVKVNVOVPZYIBLVQZBVRVSZXTJKZNOZDBPZJLVTZUULYPYRUSUUOYIBTLBYTDTWAZTFYTDTWB
+ WCWDWEWFYIUVAVRVSZUUPYIYTDTVTZUVBYIUVCUUBDTVTZYIUULUVDYHXSUUDUULUUIUUDXSU
+ ULUUMWGWHZUULUVDYOXTSODTVTDDYOWIWJWKYIYTUUBDTUUKWQWLYTDTWMWNBUVAVRFWOWNYI
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+ MXSUUFYNYPUSYIYJABCDEFGXBXCXSYHXDYHUUFXSUUJUTYJXRYDXEVGWLXFVPXSQXGYBXHZYB
+ QXIYCYGUSXSYBXGQXJUHMUVFABCDEFGHIXKXGQYBIHXLXMQXGYBXNYAYFDCQYBXTYEXRNXOXP
+ XQWL $.
$( Lemma for ~ rpnnen1 . (Contributed by Mario Carneiro, 12-May-2013.)
(Revised by NM, 13-Aug-2021.) (Proof modification is discouraged.) $)
@@ -141811,44 +141812,44 @@ Rational numbers (as a subset of complex numbers)
ffn readdcld readdcl simpl lelttr expd syl3anc mpd adantlr sylbird sylbid
peano2zd oveq1 breq1d elrab2 biimpri ad2antlr sylibr cc fvmpt2 mpan2 syld
cvv crab ssrab2 eqsstri zssre sstri a1i cmul remulcl ancoms sylan2 simpld
- btwnz zre ltdivmul rexbidva rabn0 neeq1i rabeq2i ltle impr sylan2b syldan
- ralrimiva zcnd 1cnd nncn nnne0 divdir cmpt fveq1d ovex eqid oveq1d eqtr4d
- mptex sylan9eq lenltd 3imtr3d exp31 com4l com14 rexlimdv3a pm2.01d eqlelt
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- WHUWDWGUWG $.
+ btwnz ltdivmul rexbidva rabn0 neeq1i reqabi ltle sylan2b ralrimiva syldan
+ zre impr zcnd 1cnd nncn nnne0 divdir cmpt mptex fveq1d ovex eqid sylan9eq
+ oveq1d eqtr4d 3imtr3d exp31 com4l com14 3imp rexlimdv3a pm2.01d mpbir2and
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+ HUWGWGUWD $.
$( Lemma for ~ rpnnen1 . (Contributed by Mario Carneiro, 12-May-2013.)
(Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) $)
@@ -305636,33 +305637,33 @@ arbitrary neighborhoods (such as "locally compact", which is actually
dissnref $p |- ( ( X e. V /\ U. Y = X ) -> C Ref Y ) $=
( vy wcel cuni wceq wa cv wss wrex simpr simplr r19.29a cvv eqid cref wbr
wral unisngl eqtrdi simprr snssd eqsstrd simp-4r eleqtrrd eluni2 reximddv
- csn sylib abeq2i biimpi adantl ralrimiva cpw pwexg snelpwi ad2antlr ssriv
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- WGWSIWDWHWFEVAVBVEWQRVCVFVGVHBHCFSVQVMVQTVMTVIVJVK $.
+ csn sylib eqabi biimpi adantl ralrimiva wb pwexg snelpwi ad2antlr eqeltrd
+ cpw ssriv a1i ssexd adantr isref syl mpbir2and ) EDIZFJZEKZLZCFUAUBZVMCJZ
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+ GWSIWDWHWFEVAVBVCWQRVEVFVGVHBHCFSVQVMVQTVMTVIVJVK $.
$( The set of singletons is locally finite in the discrete topology.
(Contributed by Thierry Arnoux, 9-Jan-2020.) $)
dissnlocfin $p |- ( X e. V -> C e. ( LocFin ` ~P X ) ) $=
( vz vy wcel wceq cv cin c0 wne crab cfn wa wrex csn adantl cpw ctop wral
- clocfin cfv distop eqidd snelpwi vsnid a1i nfv nfrab1 abeq2i anbi1i simpr
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- ABCEFWSXAXB $.
+ clocfin cfv distop eqidd snelpwi vsnid a1i nfrab1 nfcv eqabi anbi1i simpr
+ nfv wn simplr ineq1d disjsn2 eqtrd simp-4r neneqd pm2.65da sylib r19.29an
+ nne sneqd an32s anasss sneq rspceeqv adantll inidm eqtrdi vex eqnetrd jca
+ snnz impbida bitrid rabid velsn 3bitr4g eqrd eqeltrdi eleq2 ineq2 rabbidv
+ snfi neeq1d eleq1d anbi12d rspcev syl12anc ralrimiva unipw eqcomi unisngl
+ cuni islocfin syl3anbrc ) EDIZEUAZUBIEEJGKZHKZIZBKZXFLZMNZBCOZPIZQZHXDRZG
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+ BCEFWSXAXB $.
$}
${
@@ -306305,33 +306306,33 @@ arbitrary neighborhoods (such as "locally compact", which is actually
Carneiro, 31-Aug-2015.) $)
txbas $p |- ( ( R e. TopBases /\ S e. TopBases ) -> B e. TopBases ) $=
( vt vu vv va vb vc vd vz wcel wa cv wrex wral cxp vp ctb cin wss crn cab
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+ wceq cmpo xpeq1 xpeq2 cbvmpov rnmpo eqtri eqabi anbi12i reeanv bitr4i cop
+ basis2 exp43 imp42 opelxpi xpss12 anim12i reximi sylbir syl2an ralrimivva
+ an4s eleq1 anbi1d 2rexbidv ralxp sylibr anassrs ineq12 inxp eqtrdi sseq2d
+ anbi2d rexbidv c1st cfv c2nd rexeqi wb fvex xpex rgenw cmpt op1std op2ndd
+ cvv vex xpeq12d mpompt eqcomi eleq2 sseq1 anbi12d rexrnmptw eleq2d sseq1d
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+ EUBUBFXNHIUAGCWMXOXPXQ $.
$}
${
@@ -307408,34 +307409,34 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( x e. Y |-> ( X X. { x } ) ) e. ( S Cn ( S ^ko R ) ) ) $=
( vy vk vv vz vf cfv wcel wa cv co cima wss wceq eqid c0 ctopon cmpt cxko
csn cxp ccn wf ccnv crest ccmp cuni cpw crab cmpo crn wral cnconst2 3expa
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- elrab syl eqeltrrd wne cin cif ifnefalse ad2antlr eleq2d snss bitrdi cres
- vex df-ima simplrl ad2antrr elpwid toponuni ad5antr sseqtrrd rneqd eqtrid
- xpssres eqtrd biantrurd 3bitr2d bitr4di rabbi2dva simplrr toponss syl2anc
- rnxp bitr3d sseqin2 eqtr3d eqeltrd pm2.61dane imaeq2 mptpreima syl5ibrcom
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$.
$}
@@ -308758,10 +308759,10 @@ arbitrary neighborhoods (such as "locally compact", which is actually
ovex mp2an eqtri eqtr4di wss xkotop cin mpoexga sylan rnexg unexg restval
snex syl sylancl wo elun wb elmapg syl2anr syl5ibr ssrdv sseqtrrd sseqin2
cnf elsni sylib xkouni eqeltrd eleq1d bitrd ccmp sylan2b eqsstrd elpreima
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- mpbi simprr xkoopn ineq1 syl5ibrcom rexlimdvva jaodan fmpttd frnd tgfiss
+ rnmpo eqabi crab cres cnvresima resmptd rgenw fnmpt mp1i fveq1 fvmpt snss
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+ pm5.32da abbi2dv simprl ctopon toptopon restsn2 snfi discmp mpbi eqeltrdi
+ cpw simprr xkoopn ineq1 syl5ibrcom rexlimdvva jaodan fmpttd frnd tgfiss
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@@ -308786,15 +308787,15 @@ arbitrary neighborhoods (such as "locally compact", which is actually
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$}
@@ -308886,24 +308887,24 @@ arbitrary neighborhoods (such as "locally compact", which is actually
xkoco1cn $p |- ( ph ->
( g e. ( S Cn T ) |-> ( g o. F ) ) e. ( ( T ^ko S ) Cn ( T ^ko R ) ) ) $=
( vx vk vv vy vh ccn co cv wcel cima wa eqid ccom cmpt cxko wf ccnv crest
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+ ccmp cuni cpw crab wss cmpo crn wral cnco sylan fmpttd wceq xkobval eqabi
+ wrex ad2antrr imaeq1 imaco eqtrdi sseq1d elrab3 syl rabbidva ctop imassrn
+ wb cntop2 cnf frn sstrid imacmp sylancom simplrr xkoopn eqeltrd mptpreima
+ 3syl imaeq2 eleq1d syl5ibrcom expimpd rexlimdvva biimtrid ralrimiv ctopon
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${
@@ -308918,27 +308919,27 @@ arbitrary neighborhoods (such as "locally compact", which is actually
( g e. ( R Cn S ) |-> ( F o. g ) ) e. ( ( S ^ko R ) Cn ( T ^ko R ) ) ) $=
( vx vk vv vy vh co cv wcel cima wa syl2anc eqid ccn ccom cmpt cxko crest
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${
@@ -309636,13 +309637,13 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $=
F e. ( R Cn ( ( S tX R ) ^ko S ) ) ) $=
( vz vk vv vw wcel wa co cv wss wceq syl cvv vf vt vr ctopon cfv ctx cxko
ccn wf ccnv cima ccmp cuni cpw crab cmpo crn wral cop cmpt simplr cnmptid
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- eqid wb imaeq1 sseq1d elrab3 wfun funmpt simplrl elpwid toponuni sseqtrrd
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- syl3anc oveq2d restid eqtrd eleqtrd resttopon xpeq1d simprl xpss2 mpbird
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xpiundi eqtr3i sseq1i iunss ssin 3bitr3g anbi2d rexbidva ralrimiva eltop2
sylan2b 3syl eqeltrd imaeq2 eqtrdi syl5ibrcom expimpd rexlimdvva biimtrid
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${
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( F " S ) e. K ) $=
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$( Lemma for ~ mbfi1fseq . Verify that ` G ` describes an increasing
sequence of positive functions. (Contributed by Mario Carneiro,
@@ -402696,49 +402697,49 @@ a multiplicative function (but not completely multiplicative).
id uneq2i dmun df-suc 3eqtr4i eqtrdi mtbird rexlimi imp nosupno 3adant3
exp31 nodmon noreson syl2anc simprl3 cin dmres inss2 eqsstri inss1 sylc
a1i breq1 sltso mpan sltres syl3anc breq2d simpr resabs1d ralrimiva wor
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- simprrl sylib mtod simpll simprl2 jca simprrr weq reseq1 eqeq2d imbi12d
- cbvralvw sylibr nosupres syl113anc sylbid word nodmord ordsucss noresle
- rexlimdvaa syl23anc wfun wrel nofun funrel resdm 3syl pm2.61ian simpll1
- mtbid simpll2 nosupbnd1 simplr simpl1 sselda simpll3 mp2and impbida
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+ nosupdm eqabd simprl simplrr rspcv simprl1 sseldd soasym simprrl onelon
+ onsucb sylib mtod simpll simprl2 jca simprrr weq reseq1 eqeq2d cbvralvw
+ imbi12d sylibr nosupres rexlimdvaa sylbid word nodmord ordsucss noresle
+ syl113anc syl23anc wfun wrel nofun funrel resdm mtbid pm2.61ian simpll1
+ simpll2 nosupbnd1 simplr simpl1 sselda simpll3 sotr3 mp2and impbida
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+ VYMUWNVVHVYGUWQSVYNVYQVYPUWPHUWSYOYPXAYTUWJ $.
$}
$}
@@ -403356,47 +403357,47 @@ a multiplicative function (but not completely multiplicative).
reu5 riota1 mpbid dmeqd opeq1d sneqd eqtrd eqtrid 1oex dmsnop uneq2i dmun
uneq12d df-suc 3eqtr4ri eqtr4di reseq2d breq12d mtbird exp31 rexlimi con0
simprl3 noinfno 3adant3 ad2antrl nodmon noreson syl2anc wrel nofun funrel
- imp wfun 3syl resdm eqeltrd resdmss a1i noinfdm abeq2d simpll breq2 sylib
- simprl2 sltso mpan mpd sltres syl3anc resabs1d ralrimiva simprrl cbvralvw
- simprl simprrr weq notbid reseq1 eqeq2d imbi12d syl123anc simplrr rspcdva
- noinfres sseldd soasym onelon onsucb mtod eqnbrtrd rexlimdvaa sylbid word
- wor nodmord ordsucss noresle syl23anc breq1d mtbid simplr simpll1 simpll2
- pm2.61ian simpr simpl3 simpl1 simpl2 sselda sotr2 mp2and simpll3 impbida
- noinfbnd1 ) ELUCZEHMZILMZUDZIJUEZNOZJEUFZFIFPZQZNOZRZBUEZAUEZNORBEUFZAEUG
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$( Lemma for surreal addition properties. Show the cut properties of
surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.) $)
@@ -406820,71 +406821,71 @@ property of surreals and will be used (via surreal cuts) to prove many
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+ JYMYNYRYSYTYPUUA $.
$}
${
@@ -427149,11 +427150,11 @@ is a bijection ( 1-1 onto function) in a simple graph. (Contributed
-> F : N -1-1-onto-> I ) $=
( wcel wa cv cpr wral adantr cusgr wceq wreu wf1o co eleq2i nbusgreledg
cnbgr wb prcom eleq1i biimpi adantl prid1g eleq2 elrab2 sylanbrc sylbid
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- GVMGOVIVMDOWAPVNWABGDMVBVMCDFAHIKLVCVDVECBHGVKENVFUQ $.
+ ex biimtrid ralrimiv reqabi edgnbusgreu sylan2b ralrimiva f1ompt ) FUAO
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$}
$d I f $. $d N f n $. $d U f $.
@@ -429695,13 +429696,13 @@ a loop (see ~ uspgr1v1eop ) is a singleton of a singleton. (Contributed
/\ ( V e. Fin /\ E e. Fin ) )
-> ( sum_ v e. ( V \ { N } ) ( # ` { i e. J | v e. ( E ` i ) } )
+ ( # ` { i e. dom E | ( E ` i ) = { N } } ) ) = ( # ` J ) ) $=
- ( wcel wa cupgr cfn csn cdif cv cfv crab chash csu cdm caddc co rabeq2i
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+ ( wcel wa cupgr cfn csn cdif cv cfv crab chash csu wceq caddc co reqabi
+ cdm anbi1i anass bitri rabbia2 fveq2i a1i sumeq2dv oveq1d simpll simplr
+ simpr numedglnl syl3anc eqtrd eqtr4di ) FUASZJKSZTZKUBSEUBSTZTZKJUCZUDZ
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$d J i $. $d K v $.
$( Lemma for ~ finsumvtxdg2sstep . (Contributed by AV, 12-Dec-2021.) $)
@@ -443449,12 +443450,12 @@ then they have the same degree (i.e., the same number of adjacent
Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 2-Jan-2022.) $)
frgrwopreglem2 $p |- ( ( G e. FriendGraph /\ 1 < ( # ` V ) /\ A =/= (/) )
-> 2 <_ K ) $=
- ( cfv wbr wcel c2 cle wi wceq wa sylbi c0 wne chash clt cfrgr wex rabeq2i
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+ ( cfv wbr wcel c2 cle wi wceq wa sylbi c0 wne c1 chash cfrgr cv n0 reqabi
+ clt wex cvtxdg vdgfrgrgt2 wb breq2 fveq1i breq2i bitrdi eqcoms syl5ibrcom
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$d D x $. $d X x $. $d Y x $.
$( Lemma 3 for ~ frgrwopreg . The vertices in the sets ` A ` and ` B `
@@ -443554,13 +443555,13 @@ then they have the same degree (i.e., the same number of adjacent
-> ( ( D ` a ) = ( D ` x )
/\ ( D ` a ) =/= ( D ` b )
/\ ( D ` x ) =/= ( D ` y ) ) ) $=
- ( vz cv wcel cfv wa wceq wne rabeq2i fveqeq2 elrab2 eqtr3 expcom adantl
- wi com12 simplbiim biimtrid adantr frgrwopreglem3 ad2ant2r crab cbvrabv
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- SWCAQIVEWBHEUEURUTOUOVAVB $.
+ ( vz cv wcel cfv wa wceq wne reqabi wi elrab2 eqtr3 expcom adantl com12
+ fveqeq2 simplbiim biimtrid adantr frgrwopreglem3 ad2ant2r cbvrabv eqtri
+ imp crab ad2ant2l 3jca ) JRZCSZARZCSZUAZKRZDSZBRZDSZUAZUAVCETZVEETZUBZV
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+ WCAQIVEWBHEUKUQUROUOVAVB $.
$d A a y $. $d B a b y z $. $d E x z $.
$( Lemma 5 for ~ frgrwopreg . If ` A ` as well as ` B ` contain at least
@@ -449000,15 +449001,14 @@ Norman Megill (2007) section 1.1.3. Megill then states, "A number of
1-Apr-2006 entry_. $)
avril1 $p |- -. ( A ~P RR ( _i ` 1 ) /\ F (/) ( 0 x. 1 ) ) $=
( vy vx vz c1 ci cfv cr cpw wbr cc0 c0 wa wn cv c0r cop wcel wceq cmul co
- c1r cio cnr csn cxp cvv weq equid dfnul2 con2bii mpbi eleq1 mtbii vtocleg
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- $.
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$}
$( The law of excluded middle. Act III, Theorem 1 of Shakespeare, _Hamlet,
@@ -484812,10 +484812,10 @@ its graph has a given second element (that is, function value).
$( The range of an operation given by the maps-to notation as a subset.
(Contributed by Thierry Arnoux, 23-May-2017.) $)
rnmposs $p |- ( A. x e. A A. y e. B C e. D -> ran F C_ D ) $=
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$}
${
@@ -485836,39 +485836,39 @@ its graph has a given second element (that is, function value).
-> F : X -1-1-onto-> ( B ^m C ) ) $=
( wcel wa cres cvv wceq wf syl ad2antrr c0 vg vx wss w3a cmap co cdif csn
cv cxp cun resexg adantl simpr difexg 3ad2ant1 xpexg sylancl adantr unexg
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+ snex syl2anc adantlr ccnv cima reqabi anbi1i simprr simprll simp3 fssresd
+ elmapi simp2 simp1 ssexd elmapg mpbird eqeltrd biimpi reseq2d wfn fnresdm
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+ WFYQYRYSYT $.
$}
${
@@ -497080,69 +497080,69 @@ The sumset operation can be used for both group (additive) operations and
$( Lemma for ~ nsgqusf1o . (Contributed by Thierry Arnoux, 4-Aug-2024.) $)
nsgqusf1olem2 $p |- ( ph -> ran E = T ) $=
- ( vi crn csubg cfv cv csn co cmpt wceq wrex cab wcel wa simpr wss rabeq2i
- nsgqusf1olem1 anasss eleqtrdi sylan2b adantr eqeltrd crab sseq2 nsgmgclem
- r19.29an cnsg eleq2i nsgsubg syl subgss ad2antrr grplsmid sylancom biimpi
- nsgqus0 syl2an ssrabdv sylan2br elrabd eleqtrrdi mpteq1 eqeq2d adantl cqg
- wb rneqd cec cqs cbs eqid sselda cvv cgrp cqus a1i ovexd subgrcl eleqtrrd
- qusbas elqsi sneq oveq1d eleq1d simplr ad4antr eqtrd simpllr eqeltrrd jca
- quslsm expl reximdv2 mpd elrab sylib adantllr impbida elrnmpt elv bitr4di
- mpdan eqrdv rspcedvd abbidv rnmpt abid1 3eqtr4g eqtr4di ) AJUHZEUIUJZGAHU
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- YEYFYGYIYJYDYKIHFUUDJUDYLHYQYMYNSYO $.
+ ( vi crn csubg cfv cv csn co cmpt wceq wrex cab wcel wa wss nsgqusf1olem1
+ simpr reqabi anasss eleqtrdi sylan2b adantr r19.29an crab sseq2 nsgmgclem
+ eqeltrd cnsg eleq2i nsgsubg syl subgss ad2antrr grplsmid sylancom nsgqus0
+ biimpi syl2an sylan2br elrabd eleqtrrdi wb mpteq1 rneqd eqeq2d adantl cqg
+ ssrabdv cec cqs cbs eqid sselda cvv cgrp cqus a1i subgrcl qusbas eleqtrrd
+ ovexd elqsi sneq oveq1d eleq1d simplr ad4antr quslsm simpllr eqeltrrd jca
+ eqtrd reximdv2 mpd elrab sylib adantllr mpdan impbida elrnmpt elv bitr4di
+ expl eqrdv rspcedvd abbidv rnmpt abid1 3eqtr4g eqtr4di ) AJUHZEUIUJZGAHUK
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+ EYFYGYIYJYDYKIHFUUDJUDYLHYQYMYNSYO $.
$( Lemma for ~ nsgqusf1o . (Contributed by Thierry Arnoux, 4-Aug-2024.) $)
nsgqusf1olem3 $p |- ( ph -> ran F = S ) $=
- ( crn cv wcel csn co crab wceq wrex cvv elrnmpt elv csubg cfv wss rabeq2i
- wb wa cmpt nsgqusf1olem1 eleq2 rabbidv eqeq2d adantl wel nfv nfmpt1 nfel2
- nfrn nfan cminusg cplusg cgrp nsgsubg syl subgrcl ad4antr adantr ad3antlr
- cnsg subgss sselda simplr eqid grpasscan1 syl3anc simp-5r simp-4r cqg wbr
- ad5antr wer eqger cec erth quslsm eqeq12d bitrd biimpar w3a eqgval biimpa
- ersym simp3d syl21anc sseldd subgcl eqeltrrd adantllr ovex elrnmpti simpr
- biimpi r19.29af ovexd sneq oveq1d eqcomd elrnmptdv impbida rabbidva dfss7
- sylib eqtr2d rspcedvd anasss eleq2i eleq1 mpbird ad2antrr syl2anc nsgqus0
- nsgmgclem grplsmid eqeltrd ssrabdv sseqtrrd r19.29an bitrid bitr4id eqrdv
- jca ) AIKUGZFAIUHZUUHUIZUUIOUHZUJZNDUKZHUHZUIZOCULZUMZHGUNZUUIFUIZUUJUURV
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+ ( crn cv wcel csn co crab wceq wb cvv elrnmpt elv csubg cfv wss wa reqabi
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+ nfv nfan cminusg cplusg cgrp cnsg nsgsubg subgrcl ad4antr adantr ad3antlr
+ syl subgss sselda eqid grpasscan1 syl3anc simp-5r simp-4r cqg wbr ad5antr
+ simplr wer eqger cec quslsm eqeq12d bitrd biimpar ersym w3a eqgval biimpa
+ erth simp3d syl21anc sseldd subgcl eqeltrrd adantllr ovex elrnmpti biimpi
+ r19.29af simpr ovexd oveq1d eqcomd elrnmptdv impbida rabbidva dfss7 sylib
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$( The canonical projection homomorphism ` E ` defines a bijective
correspondence between the set ` S ` of subgroups of ` G ` containing a
@@ -516479,35 +516479,35 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
mpbir elfzp12 biimpi orcanai oveq1i eleqtrdi ss2abi abid2 ineq1i eqtr3i
1z inab 3sstr3i sstr mpan2 sylbi velpw wi wal ssab df-ex bicomi con1bii
dfclel notbii imnang bitr4i 3bitr4ri bitr2i anbi12i 3imtr4i anbi1i cneg
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+ DUJVVBVVCCDUVOTXCYR $.
$}
$d c i F $. $d j F $. $d k F $.
@@ -519947,52 +519947,52 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
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cin eqeltrd ad4antr feq1d elun1 ad3antrrr fvun1 ralrimiva sumeq2d snidg
c0 reprsum fvun2 fvsng oveq12d sselid eqeltrrd npcand r19.29ffa impbida
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+ zsscn vex snex unex elrnmpti rexbii bitr4di cbvrabv eliun 3bitr4g eqrdv
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$}
${
@@ -521487,10 +521487,10 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
$( Two ways to write the set of odd primes. (Contributed by Thierry
Arnoux, 27-Dec-2021.) $)
oddprm2 $p |- ( Prime \ { 2 } ) = ( O i^i Prime ) $=
- ( cprime c2 csn cdif cin cv wcel cdvds wbr wn wa ancom cz wb prmz rabeq2i
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- PCSTUAUBUCUJUDUJBDUEUFUG $.
+ ( cprime c2 csn cdif cin cv wcel cdvds wn wa ancom cz wb prmz reqabi baib
+ wbr syl pm5.32i bitr2i nnoddn2prmb elin 3bitr4i eqriv ) ADEFGZBDHZAIZDJZE
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+ CRSUAUBUCUJUDUJBDUEUFUG $.
hgt750leme.n $e |- ( ph -> N e. NN ) $.
${
@@ -521518,84 +521518,84 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry
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$d F e n $. $d N a c d e n $. $d O a c d e n $. $d ph a d e n $.
hgt750lema.f $e |- F = ( d e. { c e. ( NN ( repr ` 3 ) N ) | -. ( c ` a )
@@ -524500,7 +524500,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
$( First-order logic and set theory. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
bnj1436 $p |- ( x e. A -> ph ) $=
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$}
${
@@ -524593,9 +524593,9 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
$( First-order logic and set theory. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
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$}
${
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3-Jun-2011.) (New usage is discouraged.) $)
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${
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$( First-order logic and set theory. (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
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$}
${
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i e. n /\ y e. ( f ` i ) ) ) $=
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$}
${
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/\ Z e. ( f ` i ) ) ) $=
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+ wfn bitri df-rex eqabi anbi1i 3bitri w-bnj17 bnj252 anbi12i bitr4i bnj133
+ exbii 19.41v csuc c-bnj14 wceq wi com bnj1095 bnj1096 19.42 2exbii 3anass
+ nf5i 3exbii wb fndm bnj770 eleq2 3anbi2d syl 3ad2ant1 ibir impbii 3bitr2i
+ ibi ) MEHLUAZSZCJUBZIUBZUCZSZMWKWLUDZSZUEZUEZJTZKTITZCWNWPUFZJTKTITCWKKUB
+ ZSZWPUFZJTKTITWJCWQJTZUEZKTZITWTWJCKTZXEUEZXGIWJWLXBUMZABUFZKGUGZWPJWMUGZ
+ UEZXIIWJMIFJWMWOUHZUHZSZXMIFUGZXNITZWIXPMABDEFGHIJKLNOPQUIUJXQMXOSZIFUGXR
+ IMFXOUKXTXMIFJMWMWOUKULUNXRWLFSZXMUEZITXSXMIFUOYBXNIYAXLXMXLIFQUPUQVDUNUR
+ XIXBGSZXKUEZKTZXEUEXNXHYEXECYDKCYCXJABUSYDRYCXJABUTUNVDUQXLYEXMXEXKKGUOWP
+ JWMUOVAVBVCCXEKVEVCWSXFIKCWQJCJBCYCXJAJBWKVFZXBSYFWLUDDWOEHDUBVGUHVHVIJVJ
+ OVKRVLVPVMVNVBXAWRIKJCWNWPVOVQXAXDIKJXAXDXAXDCWNXAXDVRZWPCWMXBVHZYGYCXJAB
+ YHCRXBWLVSVTYHWNXCCWPWMXBWKWAWBWCZWDWHXDXACXCYGWPYIWDWEWFVQWG $.
$}
$( MARKER 1 - TODO DELETE $)
@@ -527523,7 +527523,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
lemma... of the theorem). (Contributed by Jonathan Ben-Naim,
3-Jun-2011.) (New usage is discouraged.) $)
bnj1083 $p |- ( f e. K <-> E. n ch ) $=
- ( cv wfn w3a wrex wcel wa wex df-rex abeq2i w-bnj17 bnj252 bitri 3bitr4i
+ ( cv wfn w3a wrex wcel wa wex df-rex eqabi w-bnj17 bnj252 bitri 3bitr4i
exbii ) EJZFJZKZABLZFDMZUEDNZUGOZFPUDGNCFPUGFDQUHEGIRCUJFCUIUFABSUJHUIUFA
BTUAUCUB $.
$}
@@ -528320,11 +528320,11 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
3-Jun-2011.) (New usage is discouraged.) $)
bnj1245 $p |- ( ph -> dom h C_ A ) $=
( cv wcel cdm wss w-bnj15 cres wne bnj1247 bnj1234 eleqtrdi wfn wceq wral
- wa wrex abeq2i bnj1238 bnj1196 c-bnj14 simplbi fndm bnj1241 bnj593 bnj937
- cfv syl ) ALUHZOUIZVNUJZEUKZAVNGOAEIULKUHZGUIVNGUIVRHUMVNHUMUNUDUOCEFGOIJ
- LNPQRTUAUFUGUPUQVOVQRVORUHZFUIZVNVSURZVAVQRVOWARFVOWACUHZVNVLQNVLUSCVSUTZ
- RFWAWCVARFVBLOUGVCVDVEVTWAVSEVPVTVSEUKZEIWBVFVSUKCVSUTZWDWEVARFSVCVGVSVNV
- HVIVJVKVM $.
+ cfv wrex eqabi bnj1238 bnj1196 c-bnj14 simplbi fndm bnj1241 bnj593 bnj937
+ wa syl ) ALUHZOUIZVNUJZEUKZAVNGOAEIULKUHZGUIVNGUIVRHUMVNHUMUNUDUOCEFGOIJL
+ NPQRTUAUFUGUPUQVOVQRVORUHZFUIZVNVSURZVLVQRVOWARFVOWACUHZVNVAQNVAUSCVSUTZR
+ FWAWCVLRFVBLOUGVCVDVEVTWAVSEVPVTVSEUKZEIWBVFVSUKCVSUTZWDWEVLRFSVCVGVSVNVH
+ VIVJVKVM $.
$}
${
@@ -528593,21 +528593,21 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
) ) $=
( vy cv wcel wceq vz vw w-bnj15 w3a cres wne wn w-bnj17 cfv crab wbr wral
wa wss c0 wrex bnj1232 ssrab2 cdm cin wfn c-bnj14 cop cab bnj1235 bnj1234
- biid eqid eleqtrdi abid bnj1238 bnj1196 abeq2i simplbi fndm bnj593 bnj937
- bnj1241 ssinss1 3syl sstrid bnj1253 nfrab1 nfcrii bnj1228 syl3anc bnj1309
- eqsstrid ax-5 bnj1307 hblem bnj1521 simp2 bnj1279 3adant1 bnj1280 bnj1296
- bnj982 bnj1538 necon2bi syl bnj1304 df-bnj17 mtbi imnani nne sylib ) BFUC
- ZHRZDSZIRZDSZUDZXIEUEZXKEUEZUFZUGXNXOTXMXPXHXJXLXPUHZXMXPUMXQXQARZXRXIUIZ
- XRXKUIZUFZAEUJZSZQRXRFUKUGQYBULZUDZYCAYDXQYEAYBXQXHYBBUNYBUOUFYDAYBUPXQXH
- XJXLXPXQVGZUQXQYBEBYAAEURXQEXIUSZXKUSZUTZBPXQXIXILRZVAZXSXRXIBFXRVBZUEVCZ
- JUITAYJULZUMLCUPZHVDZSZYGBUNZYIBUNXQXIDYPXQXHXJXLXPYFVEABCDYPFGHJKYMLNOYM
- VHZYPVHZVFVIYQYRLYQYJCSZYKUMYRLYQYKLCYQYKYNLCYOHVJVKVLUUAYKYJBYGUUAYJBUNZ
- YLYJUNAYJULZUUBUUCUMLCMVMVNYJXIVOVRVPVQYGYHBVSVTWHWAXQYEAQBCDEFGHIYBJKLMN
- OPYBVHZYFYEVGZWBAQUABYBFAUAYBYAAEWCWDWEWFUUEXHXJXLXPAXHAWIAUBHDAUBCDGJKLO
- AUBBCFLMWGWJZWKAUBIDUUFWKXPAWIWRWLXQYCYDWMYEXSXTTYCUGXQYEAQBCDEFGHIYBJYPX
- KYJVAXTXRXKYLUEVCZJUITAYJULUMLCUPIVDZUUGKYMLMNOPUUDYFUUEXQYEAQBCDEFGHIYBJ
- KLMNOPUUDYFUUEYCYDYLYBUTUOTXQXQYEAQBCDEFGHIYBJKLMNOPUUDYFUUEWNWOWPYSYTUUG
- VHUUHVHWQYCXSXTYAAYBEUUDWSWTXAXBXHXJXLXPXCXDXEXNXOXFXG $.
+ biid eqid eleqtrdi abid bnj1238 bnj1196 eqabi simplbi fndm bnj1241 bnj593
+ bnj937 ssinss1 3syl eqsstrid sstrid bnj1253 nfrab1 nfcrii bnj1228 syl3anc
+ ax-5 bnj1309 bnj1307 hblem bnj982 bnj1521 bnj1279 3adant1 bnj1280 bnj1296
+ simp2 bnj1538 necon2bi syl bnj1304 df-bnj17 mtbi imnani nne sylib ) BFUCZ
+ HRZDSZIRZDSZUDZXIEUEZXKEUEZUFZUGXNXOTXMXPXHXJXLXPUHZXMXPUMXQXQARZXRXIUIZX
+ RXKUIZUFZAEUJZSZQRXRFUKUGQYBULZUDZYCAYDXQYEAYBXQXHYBBUNYBUOUFYDAYBUPXQXHX
+ JXLXPXQVGZUQXQYBEBYAAEURXQEXIUSZXKUSZUTZBPXQXIXILRZVAZXSXRXIBFXRVBZUEVCZJ
+ UITAYJULZUMLCUPZHVDZSZYGBUNZYIBUNXQXIDYPXQXHXJXLXPYFVEABCDYPFGHJKYMLNOYMV
+ HZYPVHZVFVIYQYRLYQYJCSZYKUMYRLYQYKLCYQYKYNLCYOHVJVKVLUUAYKYJBYGUUAYJBUNZY
+ LYJUNAYJULZUUBUUCUMLCMVMVNYJXIVOVPVQVRYGYHBVSVTWAWBXQYEAQBCDEFGHIYBJKLMNO
+ PYBVHZYFYEVGZWCAQUABYBFAUAYBYAAEWDWEWFWGUUEXHXJXLXPAXHAWHAUBHDAUBCDGJKLOA
+ UBBCFLMWIWJZWKAUBIDUUFWKXPAWHWLWMXQYCYDWRYEXSXTTYCUGXQYEAQBCDEFGHIYBJYPXK
+ YJVAXTXRXKYLUEVCZJUITAYJULUMLCUPIVDZUUGKYMLMNOPUUDYFUUEXQYEAQBCDEFGHIYBJK
+ LMNOPUUDYFUUEYCYDYLYBUTUOTXQXQYEAQBCDEFGHIYBJKLMNOPUUDYFUUEWNWOWPYSYTUUGV
+ HUUHVHWQYCXSXTYAAYBEUUDWSWTXAXBXHXJXLXPXCXDXEXNXOXFXG $.
$}
${
@@ -528710,10 +528710,10 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
3-Jun-2011.) (New usage is discouraged.) $)
bnj1371 $p |- ( f e. H -> Fun f ) $=
( cv wcel wfun cdm csn c-bnj18 cun wceq wa wex c-bnj14 wrex bnj1436 rexex
- syl exbii sylib exsimpl wfn cfv abeq2i bnj1238 fnfun bnj31 bnj1265 bnj593
- wral bnj937 ) LUIZNUJZVQUKZEVRVQHUJZVSEVRVTVQULEUIZUMFKWAUNUOUPZUQZEURZVT
- EURVRQEURZWDVRQEFKDUIZUSZUTZWEWHLNUFVAQEWGVBVCQWCEUHVDVEVTWBEVFVCVTVSPGVT
- VQPUIZVGZVSPGVTWJWFVQVHOMVHUPDWIVOZPGWJWKUQPGUTLHTVIVJWIVQVKVLVMVNVP $.
+ syl exbii sylib exsimpl wfn wral eqabi bnj1238 fnfun bnj31 bnj1265 bnj593
+ cfv bnj937 ) LUIZNUJZVQUKZEVRVQHUJZVSEVRVTVQULEUIZUMFKWAUNUOUPZUQZEURZVTE
+ URVRQEURZWDVRQEFKDUIZUSZUTZWEWHLNUFVAQEWGVBVCQWCEUHVDVEVTWBEVFVCVTVSPGVTV
+ QPUIZVGZVSPGVTWJWFVQVOOMVOUPDWIVHZPGWJWKUQPGUTLHTVIVJWIVQVKVLVMVNVP $.
$}
${
@@ -528859,27 +528859,27 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a
wrex wbr wn wral w3a nfv nfra1 nf3an nfxfr nfiu1 nfcri nfan nf5ri bnj1521
wex w-bnj15 c0 wne crab nfe1 nfn nfcv nfrabw nfcxfr nfralw simplbi bnj835
nfne simp2bi bnj1388 rsp syl sylc bnj596 wceq bnj1373 adantl simprbi rspe
- wi syl2an abeq2i rexbii r19.42v 3bitri sylanbrc simp3bi adantr jca bnj593
- eleqtrrd df-rex sylibr dmeqi bnj1317 bnj1400 eqtri eleq2i eliun bitri cab
- cuni nfre1 nfab nfuni nfdm nfcrii bnj1397 sylbir ex ssrdv wss simpr eleq2
- biimpac reximi syl2anc rexlimiva 3imtr4i ssriv a1i eqssd ) BGINFUNZUOZGUN
- ZUPINUUJUQURZUSZMUTZBHUULUUMBHUNZUULVAZUUNUUMVAZBUUOVBZCUUPUKCUUPGCEUUPGU
- UNUUKVAZCEGUUICBUUOUURGUUIVFZUKUUSHUULGHUUIUUKVCVDVEULCGCUUQGUKBUUOGBAUUH
- LVAZUUJUUHNVGVHZGLVIZVJZGUGAUUTUVBGAGVKUUTGVKUVAGLVLVMVNGHUULGUUIUUKVOVPV
- QVNVRVSEUUNOUNZUTZVAZOQVFZUUPEUVDQVAZUVFVBZOVTUVGEETVBZUVIOETOEOECUUJUUIV
- AZUURVJOULCUVKUUROCUUQOUKBUUOOBUVCOUGAUUTUVBOAINWAZLWBWCZVBOUFUVLUVMOUVLO
- VKOLWBOLDOVTZVHZFIWDUEUVOOFIUVNODOWEWFOIWGWHWIZOWBWGWMVQVNOFLUVPVPUVAOGLU
- VPUVAOVKWJVMVNUUOOVKVQVNUVKOVKUUROVKVMVNVREBUVKTOVTZCUVKUURBEULCBUUOUKWKW
- LECUVKUURULWNZBUVQGUUIVIUVKUVQXEABDFGIJKLNOPRSTUAUBUCUDUEUFUGUHWOUVQGUUIW
- PWQWRWSUVJUVHUVFUVJUVDKVAZUVEUUKWTZGUUIVFZUVHTUVSETUVSUVTDFGIJKNOPRSTUAUB
- UCUDUHXAZWKXBEUVKUVTUWATUVRTUVSUVTUWBXCZUVTGUUIXDXFUVHTGUUIVFZUVSUVTVBZGU
- UIVFUVSUWAVBUWDOQUIXGTUWEGUUIUWBXHUVSUVTGUUIXIXJZXKUVJUUNUUKUVEEUURTECUVK
- UURULXLXMTUVTEUWCXBXPXNXOUVFOQXQXRUUPUUNOQUVEUSZVAUVGUUMUWGUUNUUMQYGZUTUW
- GMUWHUJXSOUMQUWDOUMQUIXTYAYBYCOUUNQUVEYDYEZXRXOGHUUMGMGMUWHUJGQGQUWDOYFUI
- UWDGOTGUUIYHYIWIYJWIYKYLYMYNYOYPUUMUULYQBHUUMUULUVGUUSUUPUUOUVFUUSOQUVIUV
- FUWAUUSUVHUVFYRUVHUWAUVFUVHUVSUWAUWFXCXMUVFUWAVBUVFUVTVBZGUUIVFUUSUVFUVTG
- UUIXIUWJUURGUUIUVTUVFUURUVEUUKUUNYSYTUUAYNUUBUUCUWIGUUNUUIUUKYDUUDUUEUUFU
- UG $.
+ wi syl2an eqabi rexbii r19.42v 3bitri sylanbrc adantr eleqtrrd jca bnj593
+ simp3bi df-rex sylibr cuni dmeqi bnj1317 bnj1400 eqtri eleq2i eliun bitri
+ cab nfre1 nfab nfuni nfdm nfcrii bnj1397 sylbir ssrdv simpr eleq2 biimpac
+ ex wss reximi syl2anc rexlimiva 3imtr4i ssriv a1i eqssd ) BGINFUNZUOZGUNZ
+ UPINUUJUQURZUSZMUTZBHUULUUMBHUNZUULVAZUUNUUMVAZBUUOVBZCUUPUKCUUPGCEUUPGUU
+ NUUKVAZCEGUUICBUUOUURGUUIVFZUKUUSHUULGHUUIUUKVCVDVEULCGCUUQGUKBUUOGBAUUHL
+ VAZUUJUUHNVGVHZGLVIZVJZGUGAUUTUVBGAGVKUUTGVKUVAGLVLVMVNGHUULGUUIUUKVOVPVQ
+ VNVRVSEUUNOUNZUTZVAZOQVFZUUPEUVDQVAZUVFVBZOVTUVGEETVBZUVIOETOEOECUUJUUIVA
+ ZUURVJOULCUVKUUROCUUQOUKBUUOOBUVCOUGAUUTUVBOAINWAZLWBWCZVBOUFUVLUVMOUVLOV
+ KOLWBOLDOVTZVHZFIWDUEUVOOFIUVNODOWEWFOIWGWHWIZOWBWGWMVQVNOFLUVPVPUVAOGLUV
+ PUVAOVKWJVMVNUUOOVKVQVNUVKOVKUUROVKVMVNVREBUVKTOVTZCUVKUURBEULCBUUOUKWKWL
+ ECUVKUURULWNZBUVQGUUIVIUVKUVQXEABDFGIJKLNOPRSTUAUBUCUDUEUFUGUHWOUVQGUUIWP
+ WQWRWSUVJUVHUVFUVJUVDKVAZUVEUUKWTZGUUIVFZUVHTUVSETUVSUVTDFGIJKNOPRSTUAUBU
+ CUDUHXAZWKXBEUVKUVTUWATUVRTUVSUVTUWBXCZUVTGUUIXDXFUVHTGUUIVFZUVSUVTVBZGUU
+ IVFUVSUWAVBUWDOQUIXGTUWEGUUIUWBXHUVSUVTGUUIXIXJZXKUVJUUNUUKUVEEUURTECUVKU
+ URULXPXLTUVTEUWCXBXMXNXOUVFOQXQXRUUPUUNOQUVEUSZVAUVGUUMUWGUUNUUMQXSZUTUWG
+ MUWHUJXTOUMQUWDOUMQUIYAYBYCYDOUUNQUVEYEYFZXRXOGHUUMGMGMUWHUJGQGQUWDOYGUIU
+ WDGOTGUUIYHYIWIYJWIYKYLYMYNYSYOUUMUULYTBHUUMUULUVGUUSUUPUUOUVFUUSOQUVIUVF
+ UWAUUSUVHUVFYPUVHUWAUVFUVHUVSUWAUWFXCXLUVFUWAVBUVFUVTVBZGUUIVFUUSUVFUVTGU
+ UIXIUWJUURGUUIUVTUVFUURUVEUUKUUNYQYRUUAYNUUBUUCUWIGUUNUUIUUKYEUUDUUEUUFUU
+ G $.
$}
${
@@ -530025,22 +530025,22 @@ have become an indirect lemma of the theorem in question (i.e. a lemma
bnj1211 sylib bnj1418 imim1i alimi syl bnj1142 cuni bnj1309 bnj1307 nfcii
nfuni nfcxfr nfcrii bnj1534 bnj1533 bnj1536 opeq2d fveq2d bnj1500 bnj1529
ssrab3 bnj1213 bnj1294 ax-5 3eqtr4d bnj1538 bnj836 neneqd pm2.65i c0 wrex
- nex simp1bi rabeq2i sylanbrc ne0d syl3anc bnj1209 simprbi bnj1542 bnj1525
- bnj69 mto bnj1521 bnj1541 sylbir ) HLUJZPHUKZEULZPUMZYFPHLYFUNZUOUPOUMUQE
- HURZUSANPUQUCBANPUDBCEUTCECDFUTDFDFULZNUMZYJPUMZUQDYJNHLYJUNZUOZUPZOUMZYJ
- PYMUOZUPZOUMZYKYLDYOYRODYNYQYJDGHYMNPCYJKVAZGULZYJLVBZVCZGKURZNHUKZDUGBYF
- HVAZYFNUMZYGVDZUUECUEANPVDZUUEBUDYDYEYIUUEAUCEHIJLMNOQRSTUAUBVEVFVGZVFVFC
- YTUUDYEDUGBUUFUUHYECUEAUUIYEBUDAYDYEYIUCVHVGZVFVFYMHVIDHLYJVJZVKDGHYMUUAN
- UMKUUAPUMDUUAKVAZVCZGYMDUUBUUNVLZGVMZUUAYMVAZUUNVLZGVMDUUMUUCVLZGVMUUPDUU
- CGKDCYTUUDUGVNVQUUSUUOGUUMUUBVOVPVRUUOUURGUUQUUBUUNFGHLVSVTWAWBWCUULEGUHH
- KNPUFEUHNENJWDUBEJEUHJEUHIJMOQRUAEUHHILRSWEWFWGWHWIWJZWKWLWMWNWODYKYPUQZF
- HCYTUUDUVAFHURDUGCEFUHHLNOBUUFUUHUUGYFNYHUOUPOUMUQEHURZCUEAUUIUVBBUDYDYEY
- IUVBAUCEHIJLMNOQRSTUAUBWPVFVGVFUUTWQVFDFKHUUHEHKUFWRZDCYTUUDUGVHWSZWTDYLY
- SUQZFHCYTUUDUVEFHURDUGCEFUIHLPOBUUFUUHYICUEAUUIYIBUDAYDYEYIUCVNVGVFUIULPV
- AEXAWQVFUVDWTXBDYKYLCYTUUDYKYLVDZDUGUVFFKHEFUHHKNPUFUUTWKXCXDXEXFXIUUDCDF
- KCYDKHVIZKXGVDUUDFKXHBUUFUUHYDCUEAUUIYDBUDAYDYEYIUCXJVGVFUVGCUVCVKCKYFCUU
- FUUHYFKVACBUUFUUHUEVHCBUUFUUHUEVNUUHEKHUFXKXLXMFGHKLXSXNUGXOXTXIUUHBCEHBE
- UHHNPUUJUUKBAUUIUDXPUUTXQUEABEHIJLMNOPQRSTUAUBUCUDXRYAXTYBYC $.
+ nex simp1bi reqabi sylanbrc bnj69 syl3anc bnj1209 simprbi bnj1542 bnj1525
+ ne0d mto bnj1521 bnj1541 sylbir ) HLUJZPHUKZEULZPUMZYFPHLYFUNZUOUPOUMUQEH
+ URZUSANPUQUCBANPUDBCEUTCECDFUTDFDFULZNUMZYJPUMZUQDYJNHLYJUNZUOZUPZOUMZYJP
+ YMUOZUPZOUMZYKYLDYOYRODYNYQYJDGHYMNPCYJKVAZGULZYJLVBZVCZGKURZNHUKZDUGBYFH
+ VAZYFNUMZYGVDZUUECUEANPVDZUUEBUDYDYEYIUUEAUCEHIJLMNOQRSTUAUBVEVFVGZVFVFCY
+ TUUDYEDUGBUUFUUHYECUEAUUIYEBUDAYDYEYIUCVHVGZVFVFYMHVIDHLYJVJZVKDGHYMUUANU
+ MKUUAPUMDUUAKVAZVCZGYMDUUBUUNVLZGVMZUUAYMVAZUUNVLZGVMDUUMUUCVLZGVMUUPDUUC
+ GKDCYTUUDUGVNVQUUSUUOGUUMUUBVOVPVRUUOUURGUUQUUBUUNFGHLVSVTWAWBWCUULEGUHHK
+ NPUFEUHNENJWDUBEJEUHJEUHIJMOQRUAEUHHILRSWEWFWGWHWIWJZWKWLWMWNWODYKYPUQZFH
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+ UVBAUCEHIJLMNOQRSTUAUBWPVFVGVFUUTWQVFDFKHUUHEHKUFWRZDCYTUUDUGVHWSZWTDYLYS
+ UQZFHCYTUUDUVEFHURDUGCEFUIHLPOBUUFUUHYICUEAUUIYIBUDAYDYEYIUCVNVGVFUIULPVA
+ EXAWQVFUVDWTXBDYKYLCYTUUDYKYLVDZDUGUVFFKHEFUHHKNPUFUUTWKXCXDXEXFXIUUDCDFK
+ CYDKHVIZKXGVDUUDFKXHBUUFUUHYDCUEAUUIYDBUDAYDYEYIUCXJVGVFUVGCUVCVKCKYFCUUF
+ UUHYFKVACBUUFUUHUEVHCBUUFUUHUEVNUUHEKHUFXKXLXSFGHKLXMXNUGXOXTXIUUHBCEHBEU
+ HHNPUUJUUKBAUUIUDXPUUTXQUEABEHIJLMNOPQRSTUAUBUCUDXRYAXTYBYC $.
$}
@@ -530287,17 +530287,17 @@ have become an indirect lemma of the theorem in question (i.e. a lemma
E. y A. z ( z e. y <-> E. w ( w e. x /\ A. y ph ) ) ) ) $=
( vu vv wal weq wex cfn cvv wel wa cv wcel cop nfv nfel2 nfan wi wb copab
wceq wfun wmo funopab nfa1 mof albii bitr2i cima vex eleq2w2 mpbiri imafi
- sylan2 elexd nfopab nfex issetf abeq2 exbii opabidw anbi2i bibi2i 3bitrri
- cab nfab dfima3 nfopab2 nfopab1 elequ1 opeq1 eleq1d anbi12d cbvexv1 opeq2
- anbi2d exbidv bitrid cbvabw eqtri eleq1i bitr4i sylibr sylanb expcom ) AC
- HZDCIUADHCJZEHZKLUDZDCMZEBMZWINZEJZUBZDHZCJZWKWIEDUCZUEZWLWSXAWIDUFZEHWKW
- IEDUGXBWJEWIDCACUHZUIUJUKXAWLNZWTBOZULZLPZWSXDXFKWLXAXEKPZXFKPWLXHXELPBUM
- BKLUNUOWTXEUPUQURWSWNEOZDOZQZWTPZNZEJZDVHZLPZXGXPCOZXOUDZCJWMXNUBZDHZCJWS
- CXOXNCDXMCEWNXLCWNCRCXKWTWIEDCXCUSSTUTVIVAXRXTCXNDXQVBVCXTWRCXSWQDXNWPWMX
- MWOEXLWIWNWIEDVDVEVCVFUJVCVGXFXOLXFFBMZFOZGOZQZWTPZNZFJZGVHXOFGWTXEVJYGXN
- GDYFDFYAYEDYADRDYDWTWIEDVKSTUTXNGRYGWNXIYCQZWTPZNZEJGDIZXNYFYJFEYAYEEYAER
- EYDWTWIEDVLSTYJFRFEIZYAWNYEYIFEBVMYLYDYHWTYBXIYCVNVOVPVQYKYJXMEYKYIXLWNYK
- YHXKWTYCXJXIVRVOVSVTWAWBWCWDWEWFWGWH $.
+ sylan2 elexd cab nfopab nfex nfab issetf eqab exbii opabidw anbi2i bibi2i
+ 3bitrri dfima3 nfopab2 nfopab1 elequ1 opeq1 eleq1d anbi12d cbvexv1 anbi2d
+ opeq2 exbidv bitrid cbvabw eqtri eleq1i bitr4i sylibr sylanb expcom ) ACH
+ ZDCIUADHCJZEHZKLUDZDCMZEBMZWINZEJZUBZDHZCJZWKWIEDUCZUEZWLWSXAWIDUFZEHWKWI
+ EDUGXBWJEWIDCACUHZUIUJUKXAWLNZWTBOZULZLPZWSXDXFKWLXAXEKPZXFKPWLXHXELPBUMB
+ KLUNUOWTXEUPUQURWSWNEOZDOZQZWTPZNZEJZDUSZLPZXGXPCOZXOUDZCJWMXNUBZDHZCJWSC
+ XOXNCDXMCEWNXLCWNCRCXKWTWIEDCXCUTSTVAVBVCXRXTCXNDXQVDVEXTWRCXSWQDXNWPWMXM
+ WOEXLWIWNWIEDVFVGVEVHUJVEVIXFXOLXFFBMZFOZGOZQZWTPZNZFJZGUSXOFGWTXEVJYGXNG
+ DYFDFYAYEDYADRDYDWTWIEDVKSTVAXNGRYGWNXIYCQZWTPZNZEJGDIZXNYFYJFEYAYEEYAERE
+ YDWTWIEDVLSTYJFRFEIZYAWNYEYIFEBVMYLYDYHWTYBXIYCVNVOVPVQYKYJXMEYKYIXLWNYKY
+ HXKWTYCXJXIVSVOVRVTWAWBWCWDWEWFWGWH $.
$( $j usage 'fineqvrep' avoids 'ax-pow' 'ax-rep' 'ax-inf' 'ax-inf2'; $)
$}
@@ -530308,11 +530308,11 @@ have become an indirect lemma of the theorem in question (i.e. a lemma
fineqvpow $p |- ( Fin = _V ->
E. y A. z ( A. w ( w e. z -> w e. x ) -> z e. y ) ) $=
( vv cfn cvv wceq cv wss wel wi wal wex wb cab wcel syl exbii sylib df-pw
- cpw vex eleq2w2 bitr3di mpbii elexd eqeltrrid elisset sseq1 abeq2w biimpr
- pwfi alimi eximi dfss2 imbi1i albii ) FGHZCIZAIZJZCBKZLZCMZBNZDCKDAKLDMZV
- CLZCMZBNUSVCVBOZCMZBNZVFUSBIZEIZVAJZEPZHZBNZVLUSVPGQVRUSVPVAUBZGEVAUAUSVS
- FUSVAGQZVSFQZAUCUSVAFQVTWAAFGUDVAUMUEUFUGUHBVPGUIRVQVKBVOVBECVMVNUTVAUJUK
- STVKVEBVJVDCVCVBULUNUORVEVIBVDVHCVBVGVCDUTVAUPUQURST $.
+ cpw eleq2w2 pwfi bitr3di mpbii elexd eqeltrrid elisset sseq1 eqabw biimpr
+ vex alimi eximi dfss2 imbi1i albii ) FGHZCIZAIZJZCBKZLZCMZBNZDCKDAKLDMZVC
+ LZCMZBNUSVCVBOZCMZBNZVFUSBIZEIZVAJZEPZHZBNZVLUSVPGQVRUSVPVAUBZGEVAUAUSVSF
+ USVAGQZVSFQZAUMUSVAFQVTWAAFGUCVAUDUEUFUGUHBVPGUIRVQVKBVOVBECVMVNUTVAUJUKS
+ TVKVEBVJVDCVCVBULUNUORVEVIBVDVHCVBVGVCDUTVAUPUQURST $.
$( $j usage 'fineqvpow' avoids 'ax-pow' 'ax-rep' 'ax-inf' 'ax-inf2'; $)
$}
@@ -535669,15 +535669,15 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
ntropn chmeo cpw ccvm elunii opnneip syl3anc simplr2 cbvrexvw simplr3
txopn rspe ssntr simpr1 simpr2 sseldd opeq2 ralsn anassrs dfss3 eleq1
alrimivv ralxp txtube ntrss2 mpan2 ralcom 3bitr2i ralimi bicom rexbii
- sstr r2al ralbii sylbi rabeq2i baib ad3antlr elssuni sseqtrrdi sselda
- sseq1d bibi12d syl5ibr ralimdva anim2d reximdva ssrab2 eqsstri sselid
- syl5 isclo 0elunit wrel relxp opelxp id df-3an wfn ffnd reseq1d snssd
- fnov resmpo simplll cvmlift2lem8 sylan syldan elsni eqeq1d syl5ibrcom
- 3impia mpoeq3dva 3eqtrd cnmpt1st ccom cvmlift2lem2 cnmpt21f cnmpt2res
- simp1d snex txrest mp4an oveq1i eqeltrd rspcev xpss12 restuni eleqtrd
- cncnpi expcom isopn3i eleqtrrd cnprest sylibrd embantd expimpd eleq2d
- fveq2 sylanbrc opeq2d relssdv ne0d inss2 connclo eqtr3di rabid2 iunss
- eqsstrid sseqtrdi ssrab simprbi txtopon cvmtop1 toptopon mpbir2and
+ sstr r2al ralbii reqabi baib ad3antlr elssuni sseqtrrdi sselda sseq1d
+ sylbi bibi12d syl5ibr ralimdva anim2d reximdva ssrab2 eqsstri 0elunit
+ syl5 isclo sselid wrel relxp opelxp id df-3an ffnd fnov reseq1d snssd
+ wfn resmpo simplll cvmlift2lem8 sylan syldan eqeq1d syl5ibrcom 3impia
+ mpoeq3dva 3eqtrd cnmpt1st ccom cvmlift2lem2 simp1d cnmpt21f cnmpt2res
+ elsni snex txrest oveq1i eqeltrd rspcev xpss12 restuni eleqtrd cncnpi
+ mp4an isopn3i eleqtrrd cnprest sylibrd embantd expimpd fveq2 sylanbrc
+ expcom eleq2d opeq2d relssdv ne0d inss2 connclo rabid2 iunss eqsstrid
+ eqtr3di sseqtrdi ssrab simprbi txtopon cvmtop1 toptopon mpbir2and
cncnp ) AQURURUSUTZIVAUTVBZVCUUAUUFUTZVWHVDZHQUUBZQDVIZVWFIVEUTZVFZVB
ZDVWIVGZABCDHIJLMNOPQUAUBUCUDUEUFUGUUCZAVWIVWNDVWIVHZVJZVWOAVWIRVWQAV
WITVWHVWHTVIZVKZVDZVLZRVWHTVWHVWTVLZVDVWIVXBVXCVWHVWHTVWHUUDUUGTVWHVW
@@ -535719,41 +535719,41 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by
FVWHVXNUXAWUBSFVWHVXNUWOUWPYFYKWYFWYGVXMFVXNWYGVXMWYFWXBVRZVXDVWHVYOV
DZRVJZVQZWYGWUASVWHVGZWYQWUBWUASVWHVYLWUAXQUWQWYRVXDWYPTSEFRYLWYRVYPV
YNVQZEVYTVSZSVWHVGWYPVXDWMWUAWYTSVWHVYQWYSEVYTVYNVYPUWRUWSUXBFSETRYLU
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- NVYLVXLWYPVQWYFVXNVWHVXKWWKVXNVWHVJZVYAWWKVXNWXGVWHVXNURUXGVNUXHZWFUX
- IVXLVYLWYPVXDWYPTVXKVWHGTFYEZVXAWYORXUCVWTVYOVWHVWSVXKXKXLUXJUIYPUXEY
+ XJXTYMWYNVXJVXDVXLWYPVXTVXJVXDVQAWWKWXBVXJVXTVXDVXDTGVWHUIUXCUXDUXEWY
+ NVYLVXLWYPVQWYFVXNVWHVXKWWKVXNVWHVJZVYAWWKVXNWXGVWHVXNURUXFVNUXGZWFUX
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MUXKUXLUXMUXSUXNUXOYHYJVYGGVWHVJVXSVXRVQWAGVXFVWHUIVXDTVWHUXPUXQTUKFG
- URVWHVNUXTWIWCZUXRAGVCAVCVWHVBZVWHVCVKZVDZRVJZVCGVBXUEAUYAVOZASTXUGRX
+ URVWHVNUXTWIWCZUYAAGVCAVCVWHVBZVWHVCVKZVDZRVJZVCGVBXUEAUXRVOZASTXUGRX
UGUYBAVWHXUFUYCVOVYRVWSWJZXUGVBWUDVWSXUFVBZVRAXUJRVBZVYRVWSVWHXUFUYDA
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- GWUDUEWQUFUGWWIAWUDXQXUEXUMUYAVOXCXUMXUTXUREUKURURXUTWXAXUSVRZWVNVRXU
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- VYMVWHWUJXVSUXIXVCAWUQXWDAWUDWWLXVAUYMZABCDHIJLMNOPQWUJUAUBUCUDUEUFUG
- UYNUYOUYPXWAXVKXWCXVIXWAWUKVCWUJQWUKVCUYQXNUYRUYSUYTVUAVUBXVCXVJURVYM
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- ZVXAXUGRXXCVWTXUFVWHVWSVCXKXLUXJUIYPVVHVVKAVXHVXGGURVXGVVLXUDUXRVVMUI
- VVNVXDTVWHVVOYKTVWHVXARVVPWCVVQUHVVRVWRVWIVWIVJVWOVWNDVWIVWIVVSVVTYMA
+ WWJWWKWWOVYGXVCWAVOZXXBXVRXWSWWTXFWVJVWFVUSYTVUTXVOWVJXUNQVWFIVWIHWUI
+ UAVVAXFVVBVVCVVDXRYGYHVWNXURDXUNVWIRVWKXUNVMVWMXUQQVWKXUNVWLVVEVVHUHY
+ PVVFXUKXUJXUNRXUKVWSVCVYRVWSVCVUHVVIYIUYRVVDXRVVJVXDXUHTVCVWHGVWSVCVM
+ ZVXAXUGRXXCVWTXUFVWHVWSVCXKXLUXIUIYPVVFVVKAVXHVXGGURVXGVVLXUDUYAVVMUI
+ VVQVXDTVWHVVNYKTVWHVXARVVOWCVVPUHVVRVWRVWIVWIVJVWOVWNDVWIVWIVVSVVTYMA
VWFVWIYQVFVBZIHYQVFVBZVWGVWJVWOVRVQXWIXWIXXDYRYRURURVWHVWHVWAWIAIVTVB
ZXXEAWWAXXFUBIMPVWBYMIHUAVWCYKDQVWFIVWIHVWEYTVWD $.
$}
@@ -537565,7 +537565,7 @@ codes is an increasing chain (with respect to inclusion). (Contributed
fmla0xp $p |- ( Fmla ` (/) ) = ( { (/) } X. ( _om X. _om ) ) $=
( vx vi vj vy vz c0 cv wceq com wrex cxp wcel wb cop wa eqeq2d wex adantr
elxpi adantl cfmla cfv cgoe co cvv crab cab csn fmla0 rabab goel 2rexbiia
- abeq1 0ex snid opelxpi opelxpd eleq1 syl5ibrcom rexlimivv elsni opeq1d wi
+ eqabc 0ex snid opelxpi opelxpd eleq1 syl5ibrcom rexlimivv elsni opeq1d wi
a1i simprr simpl opeq2 eqtrd jca ex 2eximdv syl6ibr syl5com sylbid impcom
r2ex exlimivv syl impbii bitri mpgbir 3eqtri ) FUAUBAGZBGZCGZUCUDZHZCIJBI
JZAUEUFWHAUGZFUHZIIKZKZABCUIWHAUJWIWLHWHWCWLLZMAWHAWLUMWHWCFWDWENZNZHZCIJ
@@ -543558,10 +543558,10 @@ Set induction (or epsilon induction)
setinds $p |- ph $=
( vz cv cvv wcel vex cab wss wi wceq setind wsbc wral dfss3 df-sbc ralbii
nfsbc1v nfcv nfralw nfim raleq sbceq1a imbi12d chvarfv sylbir sylbi sylib
- mpg eqcomi abeq2i mpbi ) BFZGHABIABGABJZGEFZUPKZUQUPHZLUPGMEEUPNURABUQOZU
- SURCFZUPHZCUQPZUTCUQUPQVCABVAOZCUQPZUTVDVBCUQABVARSVDCUOPZALVEUTLBEVEUTBV
- DBCUQBUQUAABVATUBABUQTUCUOUQMVFVEAUTVDCUOUQUDABUQUEUFDUGUHUIABUQRUJUKULUM
- UN $.
+ mpg eqcomi eqabi mpbi ) BFZGHABIABGABJZGEFZUPKZUQUPHZLUPGMEEUPNURABUQOZUS
+ URCFZUPHZCUQPZUTCUQUPQVCABVAOZCUQPZUTVDVBCUQABVARSVDCUOPZALVEUTLBEVEUTBVD
+ BCUQBUQUAABVATUBABUQTUCUOUQMVFVEAUTVDCUOUQUDABUQUEUFDUGUHUIABUQRUJUKULUMU
+ N $.
$}
${
@@ -544328,6 +544328,7 @@ Set induction (or epsilon induction)
BHIJCAABKCABLM $.
$}
+
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Surreal numbers - multiplication
@@ -545054,7 +545055,7 @@ Set induction (or epsilon induction)
( _V \ ran ( ( SSet i^i ( Trans X. _V ) ) \ ( _I u. _E ) ) ) $=
( vy vx cv wa wcel cvv csset ctrans cid cep cdif wceq wbr wex vex anbi12i
wn bitri wo anbi1i con0 wpss wtr wi wal cab cxp cin cun crn dfon2 wb elrn
- abeq1 wss brin brsset brxp mpbiran2 eltrans ioran brun ideq epel xchnxbir
+ eqabc wss brin brsset brxp mpbiran2 eltrans ioran brun ideq epel xchnxbir
orbi12i brdif dfpss2 an32 anass 3bitr4i exbii exanali eldif bitr4i mpgbir
con2bii mpbiran eqtri ) UAACZBCZUBZVTUCZDZVTWAEZUDAUEZBUFZFGHFUGZUHZIJUIZ
KZUJZKZBAUKWGWMLWFWAWMEZULBWFBWMUNWFWAWLEZQZWNWOWFWOVTWAWKMZANZWFQZAWAWKB
@@ -545418,7 +545419,7 @@ Set induction (or epsilon induction)
Fenton, 19-Feb-2013.) $)
dfiota3 $p |- ( iota x ph ) = U. U. ( { { x | ph } } i^i Singletons ) $=
( vy vz vw cab cv csn wceq cuni csingles cin wex wcel ceqsexv bitri exbii
- wa weq eqtri cio df-iota wb abeq1 wel exdistr vex sneq eqeq2d vsnex eqeq1
+ wa weq eqtri cio df-iota wb eqabc wel exdistr vex sneq eqeq2d vsnex eqeq1
eleq2 anbi12d eqcom velsn equcom bitrdi an13 bitr3i excom eluniab 3bitr4i
anbi12ci mpgbir df-sn dfsingles2 ineq12i inab 19.42v bicomi unieqi eqtr4i
abbii ) ABUAABFZCGZHZIZCFZJVNHZKLZJZJABCUBVRWAVRDGZVNIZWBEGZHZIZRZEMZDFZJ
@@ -551256,58 +551257,58 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and
csuc rspe simpll simprll fveq1i snex fr0g eqtri pwidg snssd pwexd inss2
eqtrdi elpwi sspwd ralrimivw ssralv mpan9 ssexd frsucmpt2 expcom finds2
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+ UXOJIYRUXOJQUXJUXKXPYI $.
$}
$( In the topology generated by a neighborhood base, a set is a
@@ -559107,8 +559108,8 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ).
$( A version of Russell's paradox ~ ru (see also ~ bj-ru ). (Contributed
by BJ, 12-Oct-2019.) (Proof modification is discouraged.) $)
bj-ru1 $p |- -. E. y y = { x | -. x e. x } $=
- ( cv wel wn cab wceq wb wal bj-ru0 abeq2 mtbir nex ) BCZAADEZAFGZBPABDOHA
- IABJOANKLM $.
+ ( cv wel wn cab wceq wb wal bj-ru0 eqab mtbir nex ) BCZAADEZAFGZBPABDOHAI
+ ABJOANKLM $.
$}
${
@@ -559365,11 +559366,11 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ).
$( Characterization of the elements of the singletonization of a class.
(Contributed by BJ, 6-Oct-2018.) $)
bj-elsngl $p |- ( A e. sngl B <-> E. x e. B A = { x } ) $=
- ( vy bj-csngl wcel cv wceq wa wex csn wrex dfclel df-bj-sngl abeq2i exbii
- anbi2i r19.42v bicomi 3bitri rexcom4 eqcom vsnex eqvinc exancom rexbii )
- BCEZFDGZBHZUHUGFZIZDJUIUHAGKZHZACLZIZDJZBULHZACLZDBUGMUKUODUJUNUIUNDUGDAC
- NOQPUPUIUMIZACLZDJZUSDJZACLZURUOUTDUTUOUIUMACRSPVCVAUSADCUASVBUQACUQVBUQU
- LBHUMUIIDJVBBULUBDULBAUCUDUMUIDUETSUFTT $.
+ ( vy bj-csngl wcel cv wceq wa wex csn wrex dfclel df-bj-sngl eqabi anbi2i
+ exbii r19.42v bicomi 3bitri rexcom4 eqcom vsnex eqvinc exancom rexbii ) B
+ CEZFDGZBHZUHUGFZIZDJUIUHAGKZHZACLZIZDJZBULHZACLZDBUGMUKUODUJUNUIUNDUGDACN
+ OPQUPUIUMIZACLZDJZUSDJZACLZURUOUTDUTUOUIUMACRSQVCVAUSADCUASVBUQACUQVBUQUL
+ BHUMUIIDJVBBULUBDULBAUCUDUMUIDUETSUFTT $.
$}
${
@@ -559622,10 +559623,10 @@ A comparison of the different definitions of tuples (strangely not mentioning
$( The class projection on a given component preserves unions.
(Contributed by BJ, 6-Apr-2019.) $)
bj-projun $p |- ( A Proj ( B u. C ) ) = ( ( A Proj B ) u. ( A Proj C ) ) $=
- ( vx cun bj-cproj cv wcel wo cima df-bj-proj abeq2i orbi12i elun imaundir
- csn eleq2i 3bitri 3bitr4ri eqriv ) DABCEZFZABFZACFZEZDGZUCHZUFUDHZIUFPZBA
- PZJZHZUICUJJZHZIZUFUEHUFUBHZUGULUHUNULDUCDABKLUNDUDDACKLMUFUCUDNUPUIUAUJJ
- ZHZUIUKUMEZHUOURDUBDAUAKLUQUSUIBCUJOQUIUKUMNRST $.
+ ( vx cun bj-cproj cv wcel csn cima df-bj-proj eqabi orbi12i elun imaundir
+ wo eleq2i 3bitri 3bitr4ri eqriv ) DABCEZFZABFZACFZEZDGZUCHZUFUDHZPUFIZBAI
+ ZJZHZUICUJJZHZPZUFUEHUFUBHZUGULUHUNULDUCDABKLUNDUDDACKLMUFUCUDNUPUIUAUJJZ
+ HZUIUKUMEZHUOURDUBDAUAKLUQUSUIBCUJOQUIUKUMNRST $.
$}
${
@@ -559643,12 +559644,12 @@ A comparison of the different definitions of tuples (strangely not mentioning
( A Proj ( { B } X. tag C ) ) = if ( B = A , C , (/) ) ) $=
( vx wcel csn bj-ctag cxp bj-cproj wceq c0 cif wi wn wa cima cab eleq2d
cv elsng bj-xpima2sn syl6bir imp abbidv df-bj-proj bj-taginv 3eqtr4g noel
- ex abeq2i elsni bj-xpima1sn nsyl5 bitrid mtbiri eq0rdv ifval sylanblrc wb
- eqcom ifbi ax-mp eqtrdi ) ADFZABGZCHZIZJZABKZCLMZBAKZCLMZVEVJVICKZNVJOZVI
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- VTEVIWDUKVOVSLVRWCVJVSLKABULVFVGAUMUNSUOUPUQVJVICLURUSVJVLUTVKVMKABVAVJVL
- CLVBVCVD $.
+ ex eqabi elsni bj-xpima1sn nsyl5 bitrid mtbiri ifval sylanblrc eqcom ifbi
+ eq0rdv wb ax-mp eqtrdi ) ADFZABGZCHZIZJZABKZCLMZBAKZCLMZVEVJVICKZNVJOZVIL
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+ LUTVCVD $.
$}
$( Symbols for Morse tuples. $)
@@ -559877,7 +559878,7 @@ sethood hypotheses (compare ~ opth ). (Contributed by BJ, 6-Oct-2018.) $)
${
$d A x $. $d V x $.
- $( Relative form of ~ abeq2 . (Contributed by BJ, 27-Dec-2023.) $)
+ $( Relative form of ~ eqab . (Contributed by BJ, 27-Dec-2023.) $)
bj-reabeq $p |- ( ( V i^i A ) = { x e. V | ph } <->
A. x e. V ( x e. A <-> ph ) ) $=
( cin crab wceq cv wcel wb wral dfrab3 eqeq2i nfcv nfab1 bj-rcleqf bibi2i
@@ -559955,8 +559956,8 @@ equivalent to the existence of binary unions and of nullary unions (the
(Contributed by BJ, 18-Jan-2025.)
(Proof modification is discouraged.) $)
bj-abex $p |- ( { x | ph } e. _V <-> E. y A. x ( x e. y <-> ph ) ) $=
- ( cab cvv wcel cv wceq wex wel wb wal isset abeq2 exbii bitri ) ABDZEFCGZ
- QHZCIBCJAKBLZCICQMSTCABRNOP $.
+ ( cab cvv wcel cv wceq wex wel wb wal isset eqab exbii bitri ) ABDZEFCGZQ
+ HZCIBCJAKBLZCICQMSTCABRNOP $.
$}
${
@@ -563719,18 +563720,18 @@ coordinates of a barycenter of two points in one dimension (complex
(Contributed by ML, 15-Jul-2020.) $)
f1omptsnlem $p |- F : A -1-1-onto-> R $=
( vy crn wf1o cv cfv wceq wi wcel wsbc cvv wb ax-mp wa wf1 wf wral eqid
- csn vsnex eqsbc1 mpbir sbcel2 csbconstg eleq2i bitri wrex abeq2i df-rex
- csb wex sylbbr 19.23bi sbcth sbcimg sbcan sbcel1v 3imtr3i sylanbr mpan2
- mpbi fmpti fvmpt2 mpdan sneq fvmpt3i eqeqan12d vex sneqbg bitrdi biimpd
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- VNWLWNQVOSVPVQVRAHCDEVSVTCDEWASWHDMWIWJRACWSWBZIXQBWCWHDABCWDEYCFWEGWFW
- HDCEWGSVG $.
+ csn vsnex eqsbc1 mpbir csb sbcel2 csbconstg eleq2i bitri wrex wex eqabi
+ df-rex sylbbr 19.23bi sbcth sbcimg sbcan sbcel1v 3imtr3i sylanbr fvmpt2
+ mpbi mpan2 fmpti mpdan fvmpt3i eqeqan12d vex sneqbg bitrdi biimpd rgen2
+ sneq dff13 mpbir2an f1f1orn cmpt cab rnmptsn rneqi 3eqtr4i f1oeq3 ) CEI
+ ZEJZCDEJZCDEUAZWIWKCDEUBAKZELZHKZELZMZWLWNMZNZHCUCACUCACDWLUEZEFWLCOZBK
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+ MWLWNQVNSVOVPVQAHCDEVSVTCDEWASWHDMWIWJRACWSWBZIXQBWCWHDABCWDEYCFWEGWFWH
+ DCEWGSVG $.
$}
$d A a u x z $.
@@ -563866,11 +563867,11 @@ coordinates of a barycenter of two points in one dimension (complex
topology when ` A ` is finite. (Contributed by ML, 14-Jul-2020.) $)
topdifinfindis $p |- ( A e. Fin -> T = { (/) , A } ) $=
( cfn wcel c0 cpr nfv cv cdif wn wceq wo cpw wa wi eleq1a syl wb pm4.71rd
- crab nfrab1 nfcxfr nfcv 0elpw mp1i pwidg jaod vex elpr a1i rabeq2i biortn
- diffi anbi2d bitr4id 3bitr4rd eqrd ) BEFZACGBHZUTAIACBAJZKEFZLVBGMZVBBMZN
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- DUMUTVFVGVIUTVCVFVGTBVBUOVCVFUNSUPUQURUS $.
+ crab nfrab1 nfcxfr nfcv 0elpw mp1i pwidg jaod vex a1i reqabi diffi biortn
+ elpr anbi2d bitr4id 3bitr4rd eqrd ) BEFZACGBHZUTAIACBAJZKEFZLVBGMZVBBMZNZ
+ NZABOZUBDVGAVHUCUDAVAUEUTVFVBVHFZVFPZVBVAFZVBCFZUTVFVIUTVDVIVEGVHFVDVIQUT
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+ ULUTVFVGVIUTVCVFVGTBVBUMVCVFUNSUPUQURUS $.
${
$d A u y $. $d A x y $. $d T u y $. $d T x y $.
@@ -563881,40 +563882,40 @@ coordinates of a barycenter of two points in one dimension (complex
( vu vy cfn wcel wn wceq wex wa w3a wsbc cvv c0 wo eleq1 wb ax-mp nfab1
ctopon cfv cpw cv csn wrex cab wss nfcv abid df-rex bitri eqid wi vsnex
cdif snelpwi syl5ibr imdistani anim2i 3impb 3anass sylibr mpbiri difinf
- nfv snfi sylan2 orcd ancoms 3impa syl rabeq2i 3ad2ant2 mpbid sbcth mpbi
- sbcimg sbc3an sbcg 3anbi1i eqsbc1 3anbi2i 3bitri 3anbi3i 3imtr3i mp3an2
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- syl6bi bitr4i syl6ib syl6 ax5e ssrd dissneq sylan nfielex adantr elfvex
- difss difexg elpwg 3syl adantl mpan2 0fin nsyl ad2antrl wne wpss inelcm
- cin vsnid disj4 necon2abii pssned neneqd jca pm4.56 sylib difeq2 eleq1d
- notbid eqeq1 orbi12d elrab2 biantrurd bitr4id dfin4 inss2 ssfi eqeltrri
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$d A x y $. $d T y $.
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icoreresf $p |- ( [,) |` ( RR X. RR ) ) : ( RR X. RR ) --> ~P RR $=
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$}
${
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24-Oct-2020.) $)
finxp0 $p |- ( U ^^ (/) ) = (/) $=
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$}
${
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$}
${
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${
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-> A. q e. A E! x e. ran F q e. x ) $=
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$}
${
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/\ ( Z C_ ran F /\ A C_ U. Z ) ) -> Z = ran F ) $=
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$}
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$}
$d g h k ph $.
@@ -583987,34 +583988,34 @@ the next (since the empty set has a finite subcover, the
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$d A a b $. $d B b $.
$( Property of a prime ideal in a commutative ring. (Contributed by Jeff
@@ -584626,7 +584627,7 @@ the next (since the empty set has a finite subcover, the
variable condition and in inference form. (Contributed by Giovanni
Mascellani, 4-Jun-2019.) $)
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@@ -585150,37 +585151,37 @@ A collection of theorems for commuting equalities (or
19-Aug-2018.) $)
ac6s6 $p |- E. f A. x e. A ( E. y ph -> ps ) $=
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${
@@ -635654,19 +635655,19 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM,
dib1dim $p |- ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( I ` ( R ` F ) )
= { g e. ( T X. E ) | E. s e. E g = <. ( s ` F ) , O >. } ) $=
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$}
${
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cdm breq2d rspc2v mpd cle sylib 3expa 3adantl2 ltned necomd neeq1d simpl2
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${
@@ -657543,53 +657544,53 @@ fixed reference functional determined by this vector (corresponding to
sticksstones2 $p |- ( ph -> F : A -1-1-> B ) $=
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+ spi frnd wfn ffnd hashfn syl cn0 adantr hashfz1 eqtrd eqcomd w3a wo cr cn
elfznn 3ad2ant3 nnred lttri2 syl2anc 3adant3 simp3 ffvelcdmd simprd simpr
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$}
${
@@ -679398,7 +679399,7 @@ is in the span of P(i)(X), so there is an R-linear combination of
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@@ -693723,7 +693724,7 @@ base set if and only if the neighborhoods (convergents) of every point
scottabf $p |- Scott { x | ph }
= { x | ( ph /\ A. y ( ps -> ( rank ` x ) C_ ( rank ` y ) ) ) } $=
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A. x E. y ( x e. y /\ A. z e. y A. f e. ~P y E. w e. y
( ~P z C_ ( y i^i w ) /\ ( z i^i U. f ) C_ U. ( f i^i ~P ~P w ) ) ) ) $=
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$}
${
@@ -708908,7 +708909,7 @@ not even needed (it can be any class). (Contributed by Glauco
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@@ -728461,25 +728462,25 @@ functions with (possibly) negative values. (Contributed by Glauco
Glauco Siliprandi, 20-Apr-2017.) $)
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$}
${
@@ -728923,76 +728924,76 @@ represent p__(t_i) in the paper. (Contributed by Glauco Siliprandi,
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$}
${
@@ -730304,37 +730305,37 @@ used to represent the final q_n in the paper (the one with n large
c2 cdiv nfcv rpred rehalfcld rexrd ccn sseqtrdi sseldd rfcnpre2 crab cuni
elssuni syl sseqtrrdi 2re a1i rpgt0d 2pos divgt0d eqbrtrd nffv nfbr fveq2
cr wceq breq1d elrabf sylanbrc eleqtrrdi nfrab1 nfcxfr wn wf fcnre adantr
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- sselda sylan syl3an1 adantlr 3adant1r anbi12d condan ex ssrd nf3an ssrab2
- eqsstri simplr simplll caddc cmul simpr1 simpr2 simpr3 stoweidlem41 simpr
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+ reqabi biimpi adantl simpld ffvelcdmd simprd halfpos mpbid lttrd ad2antrr
+ wb anim1i eldif sylibr rsp sylc lensymd cmpt nfv nfan eqid sselda syl2anc
+ nfra1 sylan syl3an1 adantlr 3adant1r anbi12d condan ex ssrd nf3an eqsstri
+ ssrab2 simplr simplll caddc cmul simpllr simpr1 simpr2 stoweidlem41 simpr
+ simpr3 stoweidlem49 r19.29a ralrimiva jca31 eleq2 3anbi2d rexbidv ralbidv
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$}
${
@@ -730444,21 +730445,21 @@ used to represent the final q_n in the paper (the one with n large
A. t e. B ( 1 - E ) < ( x ` t ) ) ) $=
( cv wcel cc0 cfv cle wbr c1 wa wral clt cmin co w3a wex wrex cfz wf cdiv
nfv cseq nfra1 nfan nfcv crab nfrabw nfcxfr nff nfralw cdif crp nfrab1 cn
- eqid adantr simprl wss rabeq2i simplbi cuni elssuni sseqtrrdi 3syl r19.26
- simpr crn ad2antll r19.21bi simprbi cmul 3adant1r cr adantlr stoweidlem51
- cmpt cvv c3 exlimdd df-rex sylibr ) ABVFZFVGVHEVFZYEVIZVJVKYGVLVJVKVMEJVN
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+ eqid adantr simprl wss simpr reqabi simplbi cuni elssuni sseqtrrdi r19.26
+ 3syl ad2antll r19.21bi simprbi cmul cmpt 3adant1r cr adantlr stoweidlem51
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+ WSXRYBYHBFYCYD $.
$}
${
@@ -730624,59 +730625,59 @@ used to represent the final q_n in the paper (the one with n large
syl sselda wceq nfcv crab nfcxfr sylibr ex syl2anc cvv unieq sseq2d ax-mp
wi mpd 3ad2ant3 nfv nfralw nfrabw nfpw nfin nf3an nfra1 3ad2ant1 nff eqid
ad2antrr caddc ccld cmptop iscld simprd wn disjr sylib r19.21bi eleqtrrdi
- cfz eldifd stoweidlem56 simpl simprll simprr rabeq2i sylanbrc jca32 rexex
- reximi2 nfrab1 elunif ssrdv crest cmpcld cmpsub ssrab2 eqsstri elpwg pweq
- rabexd mpbiri ineq1d rexeqdv imbi12d rspccva csn elinel1 elpwi vex difexg
- ssdifssd elpw elinel2 diffi 3ad2ant2 unidif0 sseq2i biimpri eldifsni rgen
- elind a1i raleq rspcev syl12anc rexlimdv3a nfre1 nfrexw nfss simp2 simp3l
- anbi12d simpld ccn cioo retop eqeltri cnfex sylancl sseqtrdi stoweidlem39
- ctg ssexd cmpo cseq nfex nfe1 simp1ll syld3an1 fcnre simplr simpr1 simpr2
- ad4ant14 feq3 biimpi anim1i eximi adantl uniexd c3 stoweidlem54 rexlimdva
- exlimdv ) AVKVDVLZUUKVMZRVEVLZVNZHUYSVOVPZVQZUYRSVFVLZVNZDVLZVGVLZVUCVRVR
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$}
${
@@ -748869,34 +748870,34 @@ those for the more general case of a piecewise smooth function (see
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dfss3 mpbir ssrab2 eqsstri unissi unipw sseqtri eqssi difssd ssexd mpbird
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$}
${
@@ -758854,34 +758855,34 @@ This is Step (b) of Lemma 115F of [Fremlin1] p. 31. (Contributed by
ovolval5lem3 $p |- inf ( Q , RR* , < ) = inf ( M , RR* , < ) $=
( vn cxr wceq wtru wss wcel cioo ccom cfv cn co a1i vw vg vm clt cinf crn
cv cuni cvol csumge0 wa cr cxp cmap wrex ssrab3 infxrcl mp1i cico crp cle
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${
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${
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${
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${
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+ ne0d reqabi simprbi suprclrnmpt fmptd fdmd ad2antrr eleqtrd cxr mnfxr a1i
+ rexrd an32s syldan mnfltd ffdmd biimpi suprubrnmpt fvmpt2d breqtrrd simpr
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+ dmex rgenw iinexd rabexd eqeltrid com cdom uzct saliincl elrestd ) AIUBUC
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$}
${
@@ -763305,14 +763306,14 @@ that every function in the sequence can have a different (partial)
( cfv cr wcel adantr va nfv cv cdm ciin cuni wss cle wbr wral wrex ssrab2
crab eqsstri a1i cuz cz uzid eleqtrrdi wceq fveq2 dmeqd csmblfn ffvelcdmd
syl eqid smfdmss iinssd sstrd cmpt crn clt csup wa c0 wne ne0d ffvelcdmda
- csalg smff adantlr iinss2 adantl sseli sseldd adantll rabeq2i suprclrnmpt
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+ csalg smff adantlr iinss2 adantl sseli sseldd adantll simprbi suprclrnmpt
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$}
${
@@ -764056,31 +764057,31 @@ that every function in the sequence can have a different (partial)
id wi nfv nfra1 nfan fvexi eluzelz2 zred ad2antlr cpnf cico co cin simpll
wf elinel1 csalg adantr ffvelcdmda smff syl2an simplr eluzelz2d cxr rexrd
cz pnfxr elinel2 icogelbd eluzd adantlr rspa syl2anc ffvelcdmd liminfval4
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$}
${
@@ -765159,8 +765160,8 @@ that every function in the sequence can have a different (partial)
19-Nov-2024.) $)
upwordisword $p |- ( A e. UpWord S -> A e. Word S ) $=
( vw vk cv cword wcel cupword eleq1 cfv c1 caddc clt wbr cc0 chash cmin
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- UGIUFUIGUHDEZUFJUJKZLRUFJMNDOUFPJUKQRSRTZUHULUACUICBDUBUCUDUE $.
+ co cfzo wral wa df-upword eqabi simplbi vtoclga ) CEZBFZGZAUGGCABHZUFAU
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$}
${
@@ -765222,15 +765223,15 @@ that every function in the sequence can have a different (partial)
1z 0lt1 oveq1d oveq2d eleq2d syl5ibr et-ltneverrefl fveq1 cvv ccat2s1p1
eqtrdi 1e0p1 fveq2i ccat2s1p2 eqtr3i breq12d mtbiri fvoveq1 biimpd jcad
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$}
${
@@ -773680,13 +773681,13 @@ unordered pairs consisting of exactly two (different) elements of the
29-Apr-2023.) $)
prprspr2 $p |- ( PrPairs ` V ) = { p e. ( Pairs ` V ) | ( # ` p ) = 2 } $=
( va vb cvv wcel cprpr cfv cv chash c2 wceq cspr crab wa cab wrex wb a1i
- c0 wne sprval abeq2d anbi1d r19.41vv fveqeq2 hashprg el2v bitr4di pm5.32i
- cpr biancomi 2rexbidva bitr3id bitrd abbidv df-rab prprval 3eqtr4rd fvprc
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- URUSVEVAZVHBTNZTVJVFWFTLWEVHBVBSWEVHBVITAMUTVCAGUTUSVD $.
+ c0 wne sprval eqabd anbi1d r19.41vv fveqeq2 hashprg el2v bitr4di biancomi
+ pm5.32i 2rexbidva bitr3id bitrd abbidv df-rab prprval 3eqtr4rd rab0 fvprc
+ cpr wn rabeqdv pm2.61i ) AEFZAGHZBIZJHKLZBAMHZNZLVEVGVIFZVHOZBPZCIZDIZUAZ
+ VGVNVOVAZLZOZDAQCAQZBPVJVFVEVLVTBVEVLVRDAQCAQZVHOZVTVEVKWAVHVEWABVIAEBCDU
+ BUCUDWBVRVHOZDAQCAQVEVTVRVHCDAAUEVEWCVSCDAAWCVSRVEVNAFVOAFOOWCVPVRVRVHVPV
+ RVHVQJHKLZVPVGVQKJUFVPWDRCDVNVOEEUGUHUIUKUJSULUMUNUOVJVMLVEVHBVIUPSAEBCDU
+ QURVEVBZVHBTNZTVJVFWFTLWEVHBUSSWEVHBVITAMUTVCAGUTURVD $.
$}
${
@@ -773696,10 +773697,10 @@ unordered pairs consisting of exactly two (different) elements of the
two fulfilling this wff. (Contributed by AV, 30-Apr-2023.) $)
prprsprreu $p |- ( V e. W -> ( E! p e. ( PrPairs ` V ) ph
<-> E! p e. ( Pairs ` V ) ( ( # ` p ) = 2 /\ ph ) ) ) $=
- ( wcel cv cprpr cfv wa cspr chash c2 wceq wreu wb prprspr2 rabeq2i df-reu
- weu a1i anbi1d anass bitrdi eubidv 3bitr4g ) BCEZDFZBGHZEZAIZDSUGBJHZEZUG
- KHLMZAIZIZDSADUHNUNDUKNUFUJUODUFUJULUMIZAIUOUFUIUPAUIUPOUFUMDUHUKBDPQTUAU
- LUMAUBUCUDADUHRUNDUKRUE $.
+ ( wcel cv cprpr cfv wa weu cspr chash c2 wceq wreu wb prprspr2 reqabi a1i
+ df-reu anbi1d anass bitrdi eubidv 3bitr4g ) BCEZDFZBGHZEZAIZDJUGBKHZEZUGL
+ HMNZAIZIZDJADUHOUNDUKOUFUJUODUFUJULUMIZAIUOUFUIUPAUIUPPUFUMDUHUKBDQRSUAUL
+ UMAUBUCUDADUHTUNDUKTUE $.
$}
${