Theorem ovig 6044

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 996	. 2 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( A e. R /\ B e. S ) )
2		eleq1 2259	. . . . . 6 |- ( x = A -> ( x e. R <-> A e. R ) )
3		eleq1 2259	. . . . . 6 |- ( y = B -> ( y e. S <-> B e. S ) )
4	2, 3	bi2anan9 606	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
5	4	3adant3 1019	. . . 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 473	. . 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 2120	. . . 4 |- ( E* z ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( x e. R /\ y e. S ) -> E* z ph ) )
10	8, 9	mpbir 146	. . 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 6043	. 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 429	1 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E*wmo 2046   e. wcel 2167  (class class class)co 5922   {coprab 5923

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-ov 5925  df-oprab 5926

This theorem is referenced by:  th3q  6699  addnnnq0  7516  mulnnnq0  7517  addsrpr  7812  mulsrpr  7813