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Theorem ovmpog 6009
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 6004 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 5875    e. cmpo 5877
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 4122  ax-pow 4175  ax-pr 4210  ax-setind 4537
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 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880
This theorem is referenced by:  ovmpo  6010  oav  6455  omv  6456  oeiv  6457  mapvalg  6658  pmvalg  6659  mulpipq2  7370  genipv  7508  genpelxp  7510  subval  8149  divvalap  8631  cnref1o  9650  modqval  10324  frecuzrdgrrn  10408  frec2uzrdg  10409  frecuzrdgrcl  10410  frecuzrdgsuc  10414  frecuzrdgrclt  10415  frecuzrdgg  10416  frecuzrdgsuctlem  10423  seq3val  10458  seqvalcd  10459  seqf  10461  seq3p1  10462  seqovcd  10463  seqp1cd  10466  exp3val  10522  bcval  10729  shftfvalg  10827  shftfval  10830  cnrecnv  10919  gcdval  11960  sqpweven  12175  2sqpwodd  12176  ennnfonelemp1  12407  nninfdclemcl  12449  nninfdclemp1  12451  ressvalsets  12524  imasex  12726  releqgg  13080  cnprcl2k  13709  xmetxp  14010  cncfval  14062  rpcxpef  14318  rplogbval  14366
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