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Theorem ovmpt2g 5982
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 2335 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  R  =  S )
4 ovmpt2g.3 . 2  |-  F  =  ( x  e.  C ,  y  e.  D  |->  R )
53, 4ovmpt2ga 5977 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 1623    e. wcel 1684  (class class class)co 5858    e. cmpt2 5860
This theorem is referenced by:  ovmpt2  5983  mapvalg  6782  pmvalg  6783  cdaval  7796  genpv  8623  shftfval  11565  symgov  14777  bcthlem1  18746  elghomlem1  21028  ov2gc  25123  ispr1  25156  cbcpcp  25162  isprj1  25163  ubos  25256  islimrs  25580  issubcv  25670  ismulcv  25681  valtar  25883  isword  25983  isnword  25986  isconc1  26006  isconc2  26007  isconc3  26008  mendmulr  27496  paddval  29987  tgrpov  30937  erngmul  30995  erngmul-rN  31003  dvamulr  31201  dvavadd  31204  dvhmulr  31276  djavalN  31325  djhval  31588
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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