Theorem ovig 5622

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 901	. 2 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( A e. R /\ B e. S ) )
2		eleq1 2100	. . . . . 6 |- ( x = A -> ( x e. R <-> A e. R ) )
3		eleq1 2100	. . . . . 6 |- ( y = B -> ( y e. S <-> B e. S ) )
4	2, 3	bi2anan9 538	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
5	4	3adant3 924	. . . 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 442	. . 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 1975	. . . 4 |- ( E* z ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( x e. R /\ y e. S ) -> E* z ph ) )
10	8, 9	mpbir 134	. . 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 5621	. 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 405	1 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   E*wmo 1901  (class class class)co 5512   {coprab 5513

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-ov 5515  df-oprab 5516

This theorem is referenced by:  th3q  6211  addnnnq0  6547  mulnnnq0  6548  addsrpr  6830  mulsrpr  6831