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Theorem ofmresval 6133
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 6003 . 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 1632    e. wcel 1696    X. cxp 4703    |` cres 4707  (class class class)co 5874    o Fcof 6092
This theorem is referenced by:  psradd  16143  dchrmul  20503  ldualvadd  29941
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-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-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-xp 4711  df-res 4717  df-iota 5235  df-fv 5279  df-ov 5877
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