Theorem fvopab3g 5569

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 2233	. . . 4 |- ( x = A -> ( x e. C <-> A e. C ) )
2		fvopab3g.2	. . . 4 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	anbi12d 470	. . 3 |- ( x = A -> ( ( x e. C /\ ph ) <-> ( A e. C /\ ps ) ) )
4		fvopab3g.3	. . . 4 |- ( y = B -> ( ps <-> ch ) )
5	4	anbi2d 461	. . 3 |- ( y = B -> ( ( A e. C /\ ps ) <-> ( A e. C /\ ch ) ) )
6	3, 5	opelopabg 4253	. 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 5322	. . . . 5 |- F Fn C
10		fnopfvb 5538	. . . . 5 |- ( ( F Fn C /\ A e. C ) -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
11	9, 10	mpan 422	. . . 4 |- ( A e. C -> ( ( F ` A ) = B <-> <. A , B >. e. F ) )
12	8	eleq2i 2237	. . . 4 |- ( <. A , B >. e. F <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } )
13	11, 12	bitrdi 195	. . 3 |- ( A e. C -> ( ( F ` A ) = B <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
14	13	adantr 274	. 2 |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> <. A , B >. e. { <. x , y >. | ( x e. C /\ ph ) } ) )
15		ibar 299	. . 3 |- ( A e. C -> ( ch <-> ( A e. C /\ ch ) ) )
16	15	adantr 274	. 2 |- ( ( A e. C /\ B e. D ) -> ( ch <-> ( A e. C /\ ch ) ) )
17	6, 14, 16	3bitr4d 219	1 |- ( ( A e. C /\ B e. D ) -> ( ( F ` A ) = B <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E!weu 2019   e. wcel 2141   <.cop 3586   {copab 4049   Fn wfn 5193   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  recmulnqg  7353