Theorem ov 5993

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 5877	. . . . 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 5516	. . . . 5 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
4	1, 3	eqtri 2198	. . . 4 |- ( A F B ) = ( { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } ` <. A , B >. )
5	4	eqeq1i 2185	. . 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 5977	. . . . 5 |- { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } Fn { <. x , y >. | ( x e. R /\ y e. S ) }
8		eleq1 2240	. . . . . . . 8 |- ( x = A -> ( x e. R <-> A e. R ) )
9	8	anbi1d 465	. . . . . . 7 |- ( x = A -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ y e. S ) ) )
10		eleq1 2240	. . . . . . . 8 |- ( y = B -> ( y e. S <-> B e. S ) )
11	10	anbi2d 464	. . . . . . 7 |- ( y = B -> ( ( A e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
12	9, 11	opelopabg 4268	. . . . . 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 177	. . . . 5 |- ( ( A e. R /\ B e. S ) -> <. A , B >. e. { <. x , y >. | ( x e. R /\ y e. S ) } )
14		fnopfvb 5557	. . . . 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 414	. . . 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 473	. . . . . 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 473	. . . . . 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 464	. . . . . 6 |- ( z = C -> ( ( ( A e. R /\ B e. S ) /\ ch ) <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
23	18, 20, 22	eloprabg 5962	. . . . 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 1326	. . . 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 188	. . 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	bitrid 192	. 2 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> ( ( A e. R /\ B e. S ) /\ th ) ) )
27	26	bianabs 611	1 |- ( ( A e. R /\ B e. S ) -> ( ( A F B ) = C <-> th ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E!weu 2026   e. wcel 2148   _Vcvv 2737   <.cop 3595   {copab 4063   Fn wfn 5211   ` cfv 5216  (class class class)co 5874   {coprab 5875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878

This theorem is referenced by: (None)