Theorem ov 5972

Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ov.1	|- C e. _V
ov.2	|- ( x = A -> ( ph <-> ps ) )
ov.3	|- ( y = B -> ( ps <-> ch ) )
ov.4	|- ( z = C -> ( ch <-> th ) )
ov.5	|- ( ( x e. R /\ y e. S ) -> E! z ph )
ov.6	|- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }

Assertion

Ref	Expression
ov	|- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   x, R, y, z   x, S, y, z   th, x, y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   ch( x, y, z)   F( x, y, z)

Proof of Theorem ov

Step	Hyp	Ref	Expression
1		df-ov 5856	. . . . 5 |- ( A F B ) = ( F ` <. A , B >. )
2		ov.6	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }
3	2	fveq1i 5497	. . . . 5 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
4	1, 3	eqtri 2191	. . . 4 |- ( A F B ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
5	4	eqeq1i 2178	. . 3 |- ( ( A F B ) = C <-> ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. ) = C )
6		ov.5	. . . . . 6 |- ( ( x e. R /\ y e. S ) -> E! z ph )
7	6	fnoprab 5956	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) }
8		eleq1 2233	. . . . . . . 8 |- ( x = A -> ( x e. R <-> A e. R ) )
9	8	anbi1d 462	. . . . . . 7 |- ( x = A -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ y e. S ) ) )
10		eleq1 2233	. . . . . . . 8 |- ( y = B -> ( y e. S <-> B e. S ) )
11	10	anbi2d 461	. . . . . . 7 |- ( y = B -> ( ( A e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
12	9, 11	opelopabg 4253	. . . . . 6 |- ( ( A e. R /\ B e. S ) -> ( <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } <-> ( A e. R /\ B e. S ) ) )
13	12	ibir 176	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } )
14		fnopfvb 5538	. . . . 5 |- ( ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) } /\ <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } ) -> ( ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. ) = C <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ) )
15	7, 13, 14	sylancr 412	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. ) = C <-> <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ) )
16		ov.1	. . . . 5 |- C e. _V
17		ov.2	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
18	9, 17	anbi12d 470	. . . . . 6 |- ( x = A -> ( ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( A e. R /\ y e. S ) /\ ps ) ) )
19		ov.3	. . . . . . 7 |- ( y = B -> ( ps <-> ch ) )
20	11, 19	anbi12d 470	. . . . . 6 |- ( y = B -> ( ( ( A e. R /\ y e. S ) /\ ps ) <-> ( ( A e. R /\ B e. S ) /\ ch ) ) )
21		ov.4	. . . . . . 7 |- ( z = C -> ( ch <-> th ) )
22	21	anbi2d 461	. . . . . 6 |- ( z = C -> ( ( ( A e. R /\ B e. S ) /\ ch ) <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
23	18, 20, 22	eloprabg 5941	. . . . 5 |- ( ( A e. R /\ B e. S /\ C e. _V ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
24	16, 23	mp3an3 1321	. . . 4 |- ( ( A e. R /\ B e. S ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
25	15, 24	bitrd 187	. . 3 |- ( ( A e. R /\ B e. S ) -> ( ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
26	5, 25	syl5bb 191	. 2 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
27	26	bianabs 606	1 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   E!weu 2019   e. wcel 2141   _Vcvv 2730   <.cop 3586   {copab 4049   Fn wfn 5193   ` cfv 5198  (class class class)co 5853   {coprab 5854

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  df-ov 5856  df-oprab 5857

This theorem is referenced by: (None)