Theorem oprssdmm 6116

Description: Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)

Hypotheses

Ref	Expression
oprssdmm.m	|- ( ( ph /\ u e. S ) -> E. v v e. u )
oprssdmm.cl	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
oprssdmm.f	|- ( ph -> Rel F )

Assertion

Ref	Expression
oprssdmm	|- ( ph -> ( S X. S ) C_ dom F )

Distinct variable groups:   u, F, v, x, y   u, S, x, y   ph, u, x, y

Allowed substitution hints:   ph( v)   S( v)

Proof of Theorem oprssdmm

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp6 6114	. . . . . . 7 |- ( z e. ( S X. S ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
2	1	biimpi 119	. . . . . 6 |- ( z e. ( S X. S ) -> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
3	2	adantl 275	. . . . 5 |- ( ( ph /\ z e. ( S X. S ) ) -> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
4	3	simpld 111	. . . 4 |- ( ( ph /\ z e. ( S X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
5	3	simprd 113	. . . . 5 |- ( ( ph /\ z e. ( S X. S ) ) -> ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) )
6		oprssdmm.f	. . . . . . . . 9 |- ( ph -> Rel F )
7	6	adantr 274	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> Rel F )
8		eleq2 2221	. . . . . . . . . 10 |- ( u = ( F ` <. x , y >. ) -> ( v e. u <-> v e. ( F ` <. x , y >. ) ) )
9	8	exbidv 1805	. . . . . . . . 9 |- ( u = ( F ` <. x , y >. ) -> ( E. v v e. u <-> E. v v e. ( F ` <. x , y >. ) ) )
10		oprssdmm.m	. . . . . . . . . . 11 |- ( ( ph /\ u e. S ) -> E. v v e. u )
11	10	ralrimiva 2530	. . . . . . . . . 10 |- ( ph -> A. u e. S E. v v e. u )
12	11	adantr 274	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> A. u e. S E. v v e. u )
13		df-ov 5824	. . . . . . . . . 10 |- ( x F y ) = ( F ` <. x , y >. )
14		oprssdmm.cl	. . . . . . . . . 10 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
15	13, 14	eqeltrrid 2245	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` <. x , y >. ) e. S )
16	9, 12, 15	rspcdva 2821	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> E. v v e. ( F ` <. x , y >. ) )
17		relelfvdm 5499	. . . . . . . . . 10 |- ( ( Rel F /\ v e. ( F ` <. x , y >. ) ) -> <. x , y >. e. dom F )
18	17	ex 114	. . . . . . . . 9 |- ( Rel F -> ( v e. ( F ` <. x , y >. ) -> <. x , y >. e. dom F ) )
19	18	exlimdv 1799	. . . . . . . 8 |- ( Rel F -> ( E. v v e. ( F ` <. x , y >. ) -> <. x , y >. e. dom F ) )
20	7, 16, 19	sylc 62	. . . . . . 7 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> <. x , y >. e. dom F )
21	20	ralrimivva 2539	. . . . . 6 |- ( ph -> A. x e. S A. y e. S <. x , y >. e. dom F )
22	21	adantr 274	. . . . 5 |- ( ( ph /\ z e. ( S X. S ) ) -> A. x e. S A. y e. S <. x , y >. e. dom F )
23		opeq1 3741	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. x , y >. = <. ( 1st ` z ) , y >. )
24	23	eleq1d 2226	. . . . . 6 |- ( x = ( 1st ` z ) -> ( <. x , y >. e. dom F <-> <. ( 1st ` z ) , y >. e. dom F ) )
25		opeq2 3742	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( 1st ` z ) , y >. = <. ( 1st ` z ) , ( 2nd ` z ) >. )
26	25	eleq1d 2226	. . . . . 6 |- ( y = ( 2nd ` z ) -> ( <. ( 1st ` z ) , y >. e. dom F <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F ) )
27	24, 26	rspc2va 2830	. . . . 5 |- ( ( ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) /\ A. x e. S A. y e. S <. x , y >. e. dom F ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F )
28	5, 22, 27	syl2anc 409	. . . 4 |- ( ( ph /\ z e. ( S X. S ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F )
29	4, 28	eqeltrd 2234	. . 3 |- ( ( ph /\ z e. ( S X. S ) ) -> z e. dom F )
30	29	ex 114	. 2 |- ( ph -> ( z e. ( S X. S ) -> z e. dom F ) )
31	30	ssrdv 3134	1 |- ( ph -> ( S X. S ) C_ dom F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   E.wex 1472   e. wcel 2128   A.wral 2435   C_ wss 3102   <.cop 3563   X. cxp 4583   dom cdm 4585   Rel wrel 4590   ` cfv 5169  (class class class)co 5821   1stc1st 6083   2ndc2nd 6084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-iota 5134  df-fun 5171  df-fv 5177  df-ov 5824  df-1st 6085  df-2nd 6086

This theorem is referenced by:  axaddf  7782  axmulf  7783