Theorem oprssdmm 6139

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 6137	. . . . . . 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 2230	. . . . . . . . . 10 |- ( u = ( F ` <. x , y >. ) -> ( v e. u <-> v e. ( F ` <. x , y >. ) ) )
9	8	exbidv 1813	. . . . . . . . 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 2539	. . . . . . . . . 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 5845	. . . . . . . . . 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 2254	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` <. x , y >. ) e. S )
16	9, 12, 15	rspcdva 2835	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> E. v v e. ( F ` <. x , y >. ) )
17		relelfvdm 5518	. . . . . . . . . 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 1807	. . . . . . . 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 2548	. . . . . 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 3758	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. x , y >. = <. ( 1st ` z ) , y >. )
24	23	eleq1d 2235	. . . . . 6 |- ( x = ( 1st ` z ) -> ( <. x , y >. e. dom F <-> <. ( 1st ` z ) , y >. e. dom F ) )
25		opeq2 3759	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( 1st ` z ) , y >. = <. ( 1st ` z ) , ( 2nd ` z ) >. )
26	25	eleq1d 2235	. . . . . 6 |- ( y = ( 2nd ` z ) -> ( <. ( 1st ` z ) , y >. e. dom F <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F ) )
27	24, 26	rspc2va 2844	. . . . 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 2243	. . 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 3148	1 |- ( ph -> ( S X. S ) C_ dom F )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   A.wral 2444   C_ wss 3116   <.cop 3579   X. cxp 4602   dom cdm 4604   Rel wrel 4609   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-1st 6108  df-2nd 6109

This theorem is referenced by:  axaddf  7809  axmulf  7810