Theorem fvopab3ig 5928

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 2526	. . . . . . . 8 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3ig.1	. . . . . . . 8 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 708	. . . . . . 7 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3ig.2	. . . . . . . 8 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 701	. . . . . . 7 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4754	. . . . . 6 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7	6	biimpar 483	. . . . 5 |- ( ( ( A e. C /\ B e. D ) /\ ( A e. C /\ ch ) ) -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
8	7	exp43 610	. . . 4 |- ( A e. C -> ( B e. D -> ( A e. C -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) ) )
9	8	pm2.43a 49	. . 3 |- ( A e. C -> ( B e. D -> ( ch -> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) ) )
10	9	imp 427	. 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 5849	. . 3 |- ( F ` A ) = ( { <. x , y >. | ( x e. C /\ ph ) } ` A )
13		funopab 5603	. . . . 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 2349	. . . . . 6 |- ( E* y ( x e. C /\ ph ) <-> ( x e. C -> E* y ph ) )
16	14, 15	mpbir 209	. . . . 5 |- E* y ( x e. C /\ ph )
17	13, 16	mpgbir 1627	. . . 4 |- Fun { <. x , y >. | ( x e. C /\ ph ) }
18		funopfv 5887	. . . 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 2507	. 2 |- ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } -> ( F ` A ) = B )
21	10, 20	syl6 33	1 |- ( ( A e. C /\ B e. D ) -> ( ch -> ( F ` A ) = B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   E*wmo 2285   <.cop 4022   {copab 4496   Fun wfun 5564   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578

This theorem is referenced by:  fvmptg  5929  fvopab6  5956  ov6g  6413