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Theorem ovmpog 6008
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
ovmpog.1  |-  ( x  =  A  ->  R  =  G )
ovmpog.2  |-  ( y  =  B  ->  G  =  S )
ovmpog.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpog  |-  ( ( 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)    G( x, y)    H( x, y)

Proof of Theorem ovmpog
StepHypRef Expression
1 ovmpog.1 . . 3  |-  ( x  =  A  ->  R  =  G )
2 ovmpog.2 . . 3  |-  ( y  =  B  ->  G  =  S )
31, 2sylan9eq 2230 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
4 ovmpog.3 . 2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
53, 4ovmpoga 6003 1  |-  ( ( A  e.  C  /\  B  e.  D  /\  S  e.  H )  ->  ( A F B )  =  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 978    = wceq 1353    e. wcel 2148  (class class class)co 5874    e. cmpo 5876
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879
This theorem is referenced by:  ovmpo  6009  oav  6454  omv  6455  oeiv  6456  mapvalg  6657  pmvalg  6658  mulpipq2  7369  genipv  7507  genpelxp  7509  subval  8148  divvalap  8630  cnref1o  9649  modqval  10323  frecuzrdgrrn  10407  frec2uzrdg  10408  frecuzrdgrcl  10409  frecuzrdgsuc  10413  frecuzrdgrclt  10414  frecuzrdgg  10415  frecuzrdgsuctlem  10422  seq3val  10457  seqvalcd  10458  seqf  10460  seq3p1  10461  seqovcd  10462  seqp1cd  10465  exp3val  10521  bcval  10728  shftfvalg  10826  shftfval  10829  cnrecnv  10918  gcdval  11959  sqpweven  12174  2sqpwodd  12175  ennnfonelemp1  12406  nninfdclemcl  12448  nninfdclemp1  12450  ressvalsets  12523  imasex  12725  releqgg  13078  cnprcl2k  13676  xmetxp  13977  cncfval  14029  rpcxpef  14285  rplogbval  14333
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