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Theorem ovmpt2g 5998
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
ovmpt2g.1  |-  ( x  =  A  ->  R  =  G )
ovmpt2g.2  |-  ( y  =  B  ->  G  =  S )
ovmpt2g.3  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
Assertion
Ref Expression
ovmpt2g  |-  ( ( 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 ovmpt2g
StepHypRef Expression
1 ovmpt2g.1 . . 3  |-  ( x  =  A  ->  R  =  G )
2 ovmpt2g.2 . . 3  |-  ( y  =  B  ->  G  =  S )
31, 2sylan9eq 2348 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
4 ovmpt2g.3 . 2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
53, 4ovmpt2ga 5993 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 934    = wceq 1632    e. wcel 1696  (class class class)co 5874    e. cmpt2 5876
This theorem is referenced by:  ovmpt2  5999  mapvalg  6798  pmvalg  6799  cdaval  7812  genpv  8639  shftfval  11581  symgov  14793  bcthlem1  18762  elghomlem1  21044  ov2gc  25226  ispr1  25259  cbcpcp  25265  isprj1  25266  ubos  25359  islimrs  25683  issubcv  25773  ismulcv  25784  valtar  25986  isword  26086  isnword  26089  isconc1  26109  isconc2  26110  isconc3  26111  mendmulr  27599  paddval  30609  tgrpov  31559  erngmul  31617  erngmul-rN  31625  dvamulr  31823  dvavadd  31826  dvhmulr  31898  djavalN  31947  djhval  32210
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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