Theorem ov 5620

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 5515	. . . . 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 5179	. . . . 5 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
4	1, 3	eqtri 2060	. . . 4 |- ( A F B ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
5	4	eqeq1i 2047	. . 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 5604	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) }
8		eleq1 2100	. . . . . . . 8 |- ( x = A -> ( x e. R <-> A e. R ) )
9	8	anbi1d 438	. . . . . . 7 |- ( x = A -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ y e. S ) ) )
10		eleq1 2100	. . . . . . . 8 |- ( y = B -> ( y e. S <-> B e. S ) )
11	10	anbi2d 437	. . . . . . 7 |- ( y = B -> ( ( A e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
12	9, 11	opelopabg 4005	. . . . . 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 166	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } )
14		fnopfvb 5215	. . . . 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 393	. . . 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 442	. . . . . 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 442	. . . . . 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 437	. . . . . 6 |- ( z = C -> ( ( ( A e. R /\ B e. S ) /\ ch ) <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
23	18, 20, 22	eloprabg 5592	. . . . 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 1221	. . . 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 177	. . 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 181	. 2 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
27	26	bianabs 543	1 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   E!weu 1900   _Vcvv 2557   <.cop 3378   {copab 3817   Fn wfn 4897   ` cfv 4902  (class class class)co 5512   {coprab 5513

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515  df-oprab 5516

This theorem is referenced by: (None)