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Theorem ofmresval 6117
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
Hypotheses
Ref Expression
ofmresval.f  |-  ( ph  ->  F  e.  A )
ofmresval.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
ofmresval  |-  ( ph  ->  ( F (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )

Proof of Theorem ofmresval
StepHypRef Expression
1 ofmresval.f . 2  |-  ( ph  ->  F  e.  A )
2 ofmresval.g . 2  |-  ( ph  ->  G  e.  B )
3 ovres 5987 . 2  |-  ( ( F  e.  A  /\  G  e.  B )  ->  ( F (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )
41, 2, 3syl2anc 642 1  |-  ( ph  ->  ( F (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691  (class class class)co 5858    o Fcof 6076
This theorem is referenced by:  psradd  16127  dchrmul  20487  ldualvadd  29319
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-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-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-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-ov 5861
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