Theorem ovigg 6199

Description: The value of an operation class abstraction. Compare ovig 6200. The condition ( x e. R /\ y e. S ) is been removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovigg.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
ovigg.4	|- E* z ph
ovigg.5	|- F = { <. <. x , y >. , z >. | ph }

Assertion

Ref	Expression
ovigg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> ( A F B ) = C ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ps, x, y, z

Allowed substitution hints:   ph( x, y, z)   F( x, y, z)   V( x, y, z)   W( x, y, z)   X( x, y, z)

Proof of Theorem ovigg

Step	Hyp	Ref	Expression
1		ovigg.1	. . 3 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
2	1	eloprabga 6165	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) )
3		df-ov 6078	. . . 4 |- ( A F B ) = ( F ` <. A , B >. )
4		ovigg.5	. . . . 5 |- F = { <. <. x , y >. , z >. | ph }
5	4	fveq1i 5691	. . . 4 |- ( F ` <. A , B >. ) = ( { <. <. x , y >. , z >. | ph } ` <. A , B >. )
6	3, 5	eqtri 2259	. . 3 |- ( A F B ) = ( { <. <. x , y >. , z >. | ph } ` <. A , B >. )
7		ovigg.4	. . . . 5 |- E* z ph
8	7	funoprab 6178	. . . 4 |- Fun { <. <. x , y >. , z >. | ph }
9		funopfv 5734	. . . 4 |- ( Fun { <. <. x , y >. , z >. | ph } -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } -> ( { <. <. x , y >. , z >. | ph } ` <. A , B >. ) = C ) )
10	8, 9	ax-mp 5	. . 3 |- ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } -> ( { <. <. x , y >. , z >. | ph } ` <. A , B >. ) = C )
11	6, 10	eqtrid 2283	. 2 |- ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } -> ( A F B ) = C )
12	2, 11	biimtrrdi 164	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ps -> ( A F B ) = C ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   E*wmo 2087   e. wcel 2209   <.cop 3708   Fun wfun 5366   ` cfv 5372  (class class class)co 6075   {coprab 6076

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079

This theorem is referenced by:  ovig  6200