Theorem ovmpt2a 5651

Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)

Hypotheses

Ref	Expression
ovmpt2ga.1	|- ( ( x = A /\ y = B ) -> R = S )
ovmpt2ga.2	|- F = ( x e. C , y e. D |-> R )
ovmpt2a.4	|- S e. _V

Assertion

Ref	Expression
ovmpt2a	|- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   x, S, y

Allowed substitution hints:   R( x, y)   F( x, y)

Proof of Theorem ovmpt2a

Step	Hyp	Ref	Expression
1		ovmpt2a.4	. 2 |- S e. _V
2		ovmpt2ga.1	. . 3 |- ( ( x = A /\ y = B ) -> R = S )
3		ovmpt2ga.2	. . 3 |- F = ( x e. C , y e. D |-> R )
4	2, 3	ovmpt2ga 5650	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
5	1, 4	mp3an3 1257	1 |- ( ( A e. C /\ B e. D ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601  (class class class)co 5532   |-> cmpt2 5534

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-in1 576  ax-in2 577  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  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  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-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  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  df-mpt2 5537

This theorem is referenced by: (None)