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Theorem fnfvof 6317
Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
fnfvof  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )

Proof of Theorem fnfvof
StepHypRef Expression
1 simpll 731 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  F  Fn  A )
2 simplr 732 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  G  Fn  A )
3 simpr 448 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  A  e.  V )
4 inidm 3550 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2437 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2437 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6314 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  (
( F  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
87anasss 629 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  o F R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    Fn wfn 5449   ` cfv 5454  (class class class)co 6081    o Fcof 6303
This theorem is referenced by:  ghmplusg  15461  lmhmplusg  16120  coe1addfv  16658  nmotri  18773  evlslem3  19935  evlslem1  19936  evl1addd  19954  evl1subd  19955  evl1muld  19956  mdegaddle  19997  ply1rem  20086  fta1glem2  20089  fta1blem  20091  plyexmo  20230  ulmdvlem1  20316  jensen  20827  dchrmulcl  21033  dchrinv  21045  sumdchr2  21054  dchr2sum  21057  lcomfsup  26747  mzpsubst  26805  mzpcong  27037  frlmvscaval  27208  frlmsslsp  27225  frlmup1  27227  frlmup2  27228  islindf4  27285  rngunsnply  27355  mamudi  27438  mamudir  27439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305
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