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Theorem ofmresval 6276
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 6145 . 2  |-  ( ( F  e.  A  /\  G  e.  B )  ->  ( F (  o F R  |`  ( A  X.  B ) ) G )  =  ( F  o F R G ) )
41, 2, 3syl2anc 643 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 1649    e. wcel 1717    X. cxp 4809    |` cres 4813  (class class class)co 6013    o Fcof 6235
This theorem is referenced by:  psradd  16366  dchrmul  20892  ldualvadd  29295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-res 4823  df-iota 5351  df-fv 5395  df-ov 6016
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