Theorem oprssdmm 6333

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 6331	. . . . . . 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 2295	. . . . . . . . . 10 |- ( u = ( F ` <. x , y >. ) -> ( v e. u <-> v e. ( F ` <. x , y >. ) ) )
9	8	exbidv 1873	. . . . . . . . 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 2605	. . . . . . . . . 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 6020	. . . . . . . . . 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 2319	. . . . . . . . 9 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( F ` <. x , y >. ) e. S )
16	9, 12, 15	rspcdva 2915	. . . . . . . 8 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> E. v v e. ( F ` <. x , y >. ) )
17		relelfvdm 5671	. . . . . . . . . 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 1867	. . . . . . . 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 2614	. . . . . 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 3862	. . . . . . 7 |- ( x = ( 1st ` z ) -> <. x , y >. = <. ( 1st ` z ) , y >. )
24	23	eleq1d 2300	. . . . . 6 |- ( x = ( 1st ` z ) -> ( <. x , y >. e. dom F <-> <. ( 1st ` z ) , y >. e. dom F ) )
25		opeq2 3863	. . . . . . 7 |- ( y = ( 2nd ` z ) -> <. ( 1st ` z ) , y >. = <. ( 1st ` z ) , ( 2nd ` z ) >. )
26	25	eleq1d 2300	. . . . . 6 |- ( y = ( 2nd ` z ) -> ( <. ( 1st ` z ) , y >. e. dom F <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. dom F ) )
27	24, 26	rspc2va 2924	. . . . 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 2308	. . 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 3233	1 |- ( ph -> ( S X. S ) C_ dom F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   C_ wss 3200   <.cop 3672   X. cxp 4723   dom cdm 4725   Rel wrel 4730   ` cfv 5326  (class class class)co 6017   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-1st 6302  df-2nd 6303

This theorem is referenced by:  axaddf  8087  axmulf  8088