Theorem ov 6780

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 6653	. . . . 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 6192	. . . . 5 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
4	1, 3	eqtri 2644	. . . 4 |- ( A F B ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
5	4	eqeq1i 2627	. . 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 6763	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) }
8		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. R <-> A e. R ) )
9	8	anbi1d 741	. . . . . . 7 |- ( x = A -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ y e. S ) ) )
10		eleq1 2689	. . . . . . . 8 |- ( y = B -> ( y e. S <-> B e. S ) )
11	10	anbi2d 740	. . . . . . 7 |- ( y = B -> ( ( A e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
12	9, 11	opelopabg 4993	. . . . . 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 257	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } )
14		fnopfvb 6237	. . . . 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 695	. . . 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 747	. . . . . 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 747	. . . . . 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 740	. . . . . 6 |- ( z = C -> ( ( ( A e. R /\ B e. S ) /\ ch ) <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
23	18, 20, 22	eloprabg 6748	. . . . 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 1413	. . . 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 268	. . 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 272	. 2 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
27	26	bianabs 924	1 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   _Vcvv 3200   <.cop 4183   {copab 4712   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654

This theorem is referenced by: (None)