Theorem ovidi 6779

Description: The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovidi.2	|- ( ( x e. R /\ y e. S ) -> E* z ph )
ovidi.3	|- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }

Assertion

Ref	Expression
ovidi	|- ( ( x e. R /\ y e. S ) -> ( ph -> ( x F y ) = z ) )

Distinct variable groups:   x, y, z   z, R   z, S

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

Proof of Theorem ovidi

Step	Hyp	Ref	Expression
1		ovidi.2	. . . 4 |- ( ( x e. R /\ y e. S ) -> E* z ph )
2		moanimv 2531	. . . 4 |- ( E* z ( ( x e. R /\ y e. S ) /\ ph ) <-> ( ( x e. R /\ y e. S ) -> E* z ph ) )
3	1, 2	mpbir 221	. . 3 |- E* z ( ( x e. R /\ y e. S ) /\ ph )
4		ovidi.3	. . 3 |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) }
5	3, 4	ovidig 6778	. 2 |- ( ( ( x e. R /\ y e. S ) /\ ph ) -> ( x F y ) = z )
6	5	ex 450	1 |- ( ( x e. R /\ y e. S ) -> ( ph -> ( x F y ) = z ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E*wmo 2471  (class class class)co 6650   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654

This theorem is referenced by:  ovmpt4g  6783  ov3  6797