Theorem fvopab3ig 6278

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 2689	. . . . . . . 8 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3ig.1	. . . . . . . 8 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 747	. . . . . . 7 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3ig.2	. . . . . . . 8 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 740	. . . . . . 7 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4993	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7	6	biimpar 502	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ ( A e. C /\ ch ) ) -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
8	7	exp43 640	. . . 4 |- ( A e. C -> ( B e. D -> ( A e. C -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) ) )
9	8	pm2.43a 54	. . 3 |- ( A e. C -> ( B e. D -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) )
10	9	imp 445	. 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 6192	. . 3 |- ( F ` A ) = ( { <. x , y >. | ( x e. C /\ ph ) } ` A )
13		funopab 5923	. . . . 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 2531	. . . . . 6 |- ( E* y ( x e. C /\ ph ) <-> ( x e. C -> E* y ph ) )
16	14, 15	mpbir 221	. . . . 5 |- E* y ( x e. C /\ ph )
17	13, 16	mpgbir 1726	. . . 4 |- Fun { <. x , y >. | ( x e. C /\ ph ) }
18		funopfv 6235	. . . 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 5	. . 3 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( { <. x , y >. | ( x e. C /\ ph ) } ` A ) = B )
20	12, 19	syl5eq 2668	. 2 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( F ` A ) = B )
21	10, 20	syl6 35	1 |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471   <.cop 4183   {copab 4712   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  fvmptg  6280  fvopab6  6310  ov6g  6798