Theorem ovig 6142

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 1020	. 2 |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( A e. R /\ B e. S ) )
2		eleq1 2294	. . . . . 6 |- ( x = A -> ( x e. R <-> A e. R ) )
3		eleq1 2294	. . . . . 6 |- ( y = B -> ( y e. S <-> B e. S ) )
4	2, 3	bi2anan9 610	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( x e. R /\ y e. S ) <-> ( A e. R /\ B e. S ) ) )
5	4	3adant3 1043	. . . 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 2155	. . . 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 6141	. 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 1004   = wceq 1397   E*wmo 2080   e. wcel 2202  (class class class)co 6017   {coprab 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021

This theorem is referenced by:  th3q  6808  addnnnq0  7668  mulnnnq0  7669  addsrpr  7964  mulsrpr  7965