Theorem fvopab3ig 5246

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Hypotheses

Ref	Expression
fvopab3ig.1	|- ( x = A -> ( ph <-> ps ) )
fvopab3ig.2	|- ( y = B -> ( ps <-> ch ) )
fvopab3ig.3	|- ( x e. C -> E* y ph )
fvopab3ig.4	|- F = { <. x , y >. | ( x e. C /\ ph ) }

Assertion

Ref	Expression
fvopab3ig	|- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   ch, x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   D( x, y)   F( x, y)

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 2100	. . . . . . . 8 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3ig.1	. . . . . . . 8 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 442	. . . . . . 7 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3ig.2	. . . . . . . 8 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 437	. . . . . . 7 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4005	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7	6	biimpar 281	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ ( A e. C /\ ch ) ) -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
8	7	exp43 354	. . . 4 |- ( A e. C -> ( B e. D -> ( A e. C -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) ) )
9	8	pm2.43a 45	. . 3 |- ( A e. C -> ( B e. D -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) )
10	9	imp 115	. 2 |- ( ( A e. C /\ B e. D ) -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
11		fvopab3ig.4	. . . 4 |- F = { <. x , y >. | ( x e. C /\ ph ) }
12	11	fveq1i 5179	. . 3 |- ( F ` A ) = ( { <. x , y >. | ( x e. C /\ ph ) } ` A )
13		funopab 4935	. . . . 5 |- ( Fun { <. x , y >. | ( x e. C /\ ph ) } <-> A. x E* y ( x e. C /\ ph ) )
14		fvopab3ig.3	. . . . . 6 |- ( x e. C -> E* y ph )
15		moanimv 1975	. . . . . 6 |- ( E* y ( x e. C /\ ph ) <-> ( x e. C -> E* y ph ) )
16	14, 15	mpbir 134	. . . . 5 |- E* y ( x e. C /\ ph )
17	13, 16	mpgbir 1342	. . . 4 |- Fun { <. x , y >. | ( x e. C /\ ph ) }
18		funopfv 5213	. . . 4 |- ( Fun { <. x , y >. | ( x e. C /\ ph ) } -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( { <. x , y >. | ( x e. C /\ ph ) } ` A ) = B ) )
19	17, 18	ax-mp 7	. . 3 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( { <. x , y >. | ( x e. C /\ ph ) } ` A ) = B )
20	12, 19	syl5eq 2084	. 2 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( F ` A ) = B )
21	10, 20	syl6 29	1 |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E*wmo 1901   <.cop 3378   {copab 3817   Fun wfun 4896   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by:  fvmptg  5248  fvopab6  5264  ov6g  5638