Theorem ovmpt2ga 5661

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

Hypotheses

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

Assertion

Ref	Expression
ovmpt2ga	|- ( ( A e. C /\ B e. D /\ S e. H ) -> ( 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)   H( x, y)

Proof of Theorem ovmpt2ga

Step	Hyp	Ref	Expression
1		elex 2611	. 2 |- ( S e. H -> S e. _V )
2		ovmpt2ga.2	. . . 4 |- F = ( x e. C , y e. D |-> R )
3	2	a1i 9	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> F = ( x e. C , y e. D |-> R ) )
4		ovmpt2ga.1	. . . 4 |- ( ( x = A /\ y = B ) -> R = S )
5	4	adantl 271	. . 3 |- ( ( ( A e. C /\ B e. D /\ S e. _V ) /\ ( x = A /\ y = B ) ) -> R = S )
6		simp1 939	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> A e. C )
7		simp2 940	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> B e. D )
8		simp3 941	. . 3 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> S e. _V )
9	3, 5, 6, 7, 8	ovmpt2d 5659	. 2 |- ( ( A e. C /\ B e. D /\ S e. _V ) -> ( A F B ) = S )
10	1, 9	syl3an3 1205	1 |- ( ( A e. C /\ B e. D /\ S e. H ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434   _Vcvv 2602  (class class class)co 5543   |-> cmpt2 5545

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548

This theorem is referenced by:  ovmpt2a  5662  ovmpt2g  5666  elovmpt2  5732  offval  5750  offval3  5792  fzoval  9235  eucalgval2  10579