Theorem fvopab3g 5675

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 2270	. . . 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 4332	. 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 5420	. . . . 5 |- F Fn C
10		fnopfvb 5643	. . . . 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 2274	. . . 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 2055   e. wcel 2178   <.cop 3646   {copab 4120   Fn wfn 5285   ` cfv 5290

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  recmulnqg  7539