Theorem oprssdmm 6166

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 6164	. . . . . . 7 |- ( z e. ( S X. S ) <-> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
2	1	biimpi 120	. . . . . 6 |- ( z e. ( S X. S ) -> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
3	2	adantl 277	. . . . 5 |- ( ( ph /\ z e. ( S X. S ) ) -> ( z = <. ( 1st ` z ) , ( 2nd ` z ) >. /\ ( ( 1st ` z ) e. S /\ ( 2nd ` z ) e. S ) ) )
4	3	simpld 112	. . . 4 |- ( ( ph /\ z e. ( S X. S ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
5	3	simprd 114	. . . . 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 276	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> Rel F )
8		eleq2 2241	. . . . . . . . . 10 |- ( u = ( F ` <. x , y >. ) -> ( v e. u <-> v e. ( F ` <. x , y >. ) ) )
9	8	exbidv 1825	. . . . . . . . 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 2550	. . . . . . . . . 10 |- ( ph -> A. u e. S E. v v e. u )
12	11	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> A. u e. S E. v v e. u )
13		df-ov 5872	. . . . . . . . . 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 2265	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` <. x , y >. ) e. S )
16	9, 12, 15	rspcdva 2846	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> E. v v e. ( F ` <. x , y >. ) )
17		relelfvdm 5543	. . . . . . . . . 10 |- ( ( Rel F /\ v e. ( F ` <. x , y >. ) ) -> <. x , y >. e. dom F )
18	17	ex 115	. . . . . . . . 9 |- ( Rel F -> ( v e. ( F ` <. x , y >. ) -> <. x , y >. e. dom F ) )
19	18	exlimdv 1819	. . . . . . . 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 2559	. . . . . 6 |- ( ph -> A. x e. S A. y e. S <. x , y >. e. dom F )
22	21	adantr 276	. . . . 5 |- ( ( ph /\ z e. ( S X. S ) ) -> A. x e. S A. y e. S <. x , y >. e. dom F )
23		opeq1 3776	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. x , y >. = <. ( 1st ` z ) , y >. )
24	23	eleq1d 2246	. . . . . 6 |- ( x = ( 1st ` z ) -> ( <. x , y >. e. dom F <-> <. ( 1st ` z ) , y >. e. dom F ) )
25		opeq2 3777	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( 1st ` z ) , y >. = <. ( 1st ` z ) , ( 2nd ` z ) >. )
26	25	eleq1d 2246	. . . . . 6 |- ( y = ( 2nd ` z ) -> ( <. ( 1st ` z ) , y >. e. dom F <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F ) )
27	24, 26	rspc2va 2855	. . . . 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 411	. . . 4 |- ( ( ph /\ z e. ( S X. S ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F )
29	4, 28	eqeltrd 2254	. . 3 |- ( ( ph /\ z e. ( S X. S ) ) -> z e. dom F )
30	29	ex 115	. 2 |- ( ph -> ( z e. ( S X. S ) -> z e. dom F ) )
31	30	ssrdv 3161	1 |- ( ph -> ( S X. S ) C_ dom F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   C_ wss 3129   <.cop 3594   X. cxp 4621   dom cdm 4623   Rel wrel 4628   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fv 5220  df-ov 5872  df-1st 6135  df-2nd 6136

This theorem is referenced by:  axaddf  7858  axmulf  7859