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Theorem ovmpog 5871
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 2168 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
4 ovmpog.3 . 2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
53, 4ovmpoga 5866 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 945    = wceq 1314    e. wcel 1463  (class class class)co 5740    e. cmpo 5742
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  ovmpo  5872  oav  6316  omv  6317  oeiv  6318  mapvalg  6518  pmvalg  6519  mulpipq2  7143  genipv  7281  genpelxp  7283  subval  7918  divvalap  8397  cnref1o  9392  modqval  10048  frecuzrdgrrn  10132  frec2uzrdg  10133  frecuzrdgrcl  10134  frecuzrdgsuc  10138  frecuzrdgrclt  10139  frecuzrdgg  10140  frecuzrdgsuctlem  10147  seq3val  10182  seqvalcd  10183  seqf  10185  seq3p1  10186  seqovcd  10187  seqp1cd  10190  exp3val  10246  bcval  10446  shftfvalg  10541  shftfval  10544  cnrecnv  10633  gcdval  11555  sqpweven  11759  2sqpwodd  11760  ennnfonelemp1  11825  cnprcl2k  12281  xmetxp  12582  cncfval  12634
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