Theorem ov 5640

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 5535	. . . . 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 5199	. . . . 5 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
4	1, 3	eqtri 2101	. . . 4 |- ( A F B ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
5	4	eqeq1i 2088	. . 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 5624	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) }
8		eleq1 2141	. . . . . . . 8 |- ( x = A -> ( x e. R <-> A e. R ) )
9	8	anbi1d 452	. . . . . . 7 |- ( x = A -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ y e. S ) ) )
10		eleq1 2141	. . . . . . . 8 |- ( y = B -> ( y e. S <-> B e. S ) )
11	10	anbi2d 451	. . . . . . 7 |- ( y = B -> ( ( A e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
12	9, 11	opelopabg 4023	. . . . . 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 175	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } )
14		fnopfvb 5236	. . . . 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 405	. . . 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 456	. . . . . 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 456	. . . . . 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 451	. . . . . 6 |- ( z = C -> ( ( ( A e. R /\ B e. S ) /\ ch ) <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
23	18, 20, 22	eloprabg 5612	. . . . 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 1257	. . . 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 186	. . 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 190	. 2 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
27	26	bianabs 575	1 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E!weu 1941   _Vcvv 2601   <.cop 3401   {copab 3838   Fn wfn 4917   ` cfv 4922  (class class class)co 5532   {coprab 5533

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535  df-oprab 5536

This theorem is referenced by: (None)