Theorem fvopab3g 5592

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 2240	. . . 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 4270	. 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 5342	. . . . 5 |- F Fn C
10		fnopfvb 5560	. . . . 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 2244	. . . 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 1353   E!weu 2026   e. wcel 2148   <.cop 3597   {copab 4065   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  recmulnqg  7393