Theorem fvopab3g 5654

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
fvopab3g.2	|- ( x = A -> ( ph <-> ps ) )
fvopab3g.3	|- ( y = B -> ( ps <-> ch ) )
fvopab3g.4	|- ( x e. C -> E! y ph )
fvopab3g.5	|- F = { <. x , y >. | ( x e. C /\ ph ) }

Assertion

Ref	Expression
fvopab3g	|- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) )

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 fvopab3g

Step	Hyp	Ref	Expression
1		eleq1 2268	. . . 4 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3g.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 473	. . 3 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3g.3	. . . 4 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 464	. . 3 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4315	. 2 |- ( ( A e. C /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } <-> ( A e. C /\ ch ) ) )
7		fvopab3g.4	. . . . . 6 |- ( x e. C -> E! y ph )
8		fvopab3g.5	. . . . . 6 |- F = { <. x , y >. | ( x e. C /\ ph ) }
9	7, 8	fnopab 5402	. . . . 5 |- F Fn C
10		fnopfvb 5622	. . . . 5 |- ( ( F Fn C /\ A e. C ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
11	9, 10	mpan 424	. . . 4 |- ( A e. C -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
12	8	eleq2i 2272	. . . 4 |- ( <. A , B >. e. F <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
13	11, 12	bitrdi 196	. . 3 |- ( A e. C -> ( ( F ` A ) = B <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
14	13	adantr 276	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
15		ibar 301	. . 3 |- ( A e. C -> ( ch <-> ( A e. C /\ ch ) ) )
16	15	adantr 276	. 2 |- ( ( A e. C /\ B e. D ) -> ( ch <-> ( A e. C /\ ch ) ) )
17	6, 14, 16	3bitr4d 220	1 |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E!weu 2054   e. wcel 2176   <.cop 3636   {copab 4105   Fn wfn 5267   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  recmulnqg  7506