Theorem ovig 5642

Description: The value of an operation class abstraction (weak version). (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovig.1	|- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
ovig.2	|- ( ( x e. R /\ y e. S ) -> E* z ph )
ovig.3	|- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }

Assertion

Ref	Expression
ovig	|- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )

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

Allowed substitution hints:   ph( x, y, z)   D( x, y, z)   F( x, y, z)

Proof of Theorem ovig

Step	Hyp	Ref	Expression
1		3simpa 935	. 2 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( A e. R /\ B e. S ) )
2		eleq1 2141	. . . . . 6 |- ( x = A -> ( x e. R <-> A e. R ) )
3		eleq1 2141	. . . . . 6 |- ( y = B -> ( y e. S <-> B e. S ) )
4	2, 3	bi2anan9 570	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
5	4	3adant3 958	. . . 4 |- ( ( x = A /\ y = B /\ z = C ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
6		ovig.1	. . . 4 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) )
7	5, 6	anbi12d 456	. . 3 |- ( ( x = A /\ y = B /\ z = C ) -> ( ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( A e. R /\ B e. S ) /\ ps ) ) )
8		ovig.2	. . . 4 |- ( ( x e. R /\ y e. S ) -> E* z ph )
9		moanimv 2016	. . . 4 |- ( E* z ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( x e. R /\ y e. S ) -> E* z ph ) )
10	8, 9	mpbir 144	. . 3 |- E* z ( ( x e. R /\ y e. S ) /\ ph )
11		ovig.3	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }
12	7, 10, 11	ovigg 5641	. 2 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ( ( A e. R /\ B e. S ) /\ ps ) -> ( A F B ) = C ) )
13	1, 12	mpand 419	1 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E*wmo 1942  (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-fv 4930  df-ov 5535  df-oprab 5536

This theorem is referenced by:  th3q  6234  addnnnq0  6639  mulnnnq0  6640  addsrpr  6922  mulsrpr  6923