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Theorem ovmpoa 5894
Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.)
Hypotheses
Ref Expression
ovmpoga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
ovmpoga.2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
ovmpoa.4  |-  S  e. 
_V
Assertion
Ref Expression
ovmpoa  |-  ( ( 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 ovmpoa
StepHypRef Expression
1 ovmpoa.4 . 2  |-  S  e. 
_V
2 ovmpoga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
3 ovmpoga.2 . . 3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
42, 3ovmpoga 5893 . 2  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  _V )  ->  ( A F B )  =  S )
51, 4mp3an3 1304 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2681  (class class class)co 5767    e. cmpo 5769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772
This theorem is referenced by: (None)
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