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Theorem fnfvof 6535
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  oF R G ) `
 X )  =  ( ( F `  X ) R ( G `  X ) ) )

Proof of Theorem fnfvof
StepHypRef Expression
1 simpll 752 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  F  Fn  A )
2 simplr 754 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  G  Fn  A )
3 simpr 459 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  A  e.  V )
4 inidm 3648 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2403 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2403 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6530 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  (
( F  oF R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
87anasss 645 1  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A  e.  V  /\  X  e.  A ) )  -> 
( ( F  oF R G ) `
 X )  =  ( ( F `  X ) R ( G `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    Fn wfn 5564   ` cfv 5569  (class class class)co 6278    oFcof 6519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521
This theorem is referenced by:  ghmplusg  17176  lcomfsupOLD  17869  lcomfsupp  17870  lmhmplusg  18010  evlslem3  18503  evlslem1  18504  coe1addfv  18626  evl1addd  18697  evl1subd  18698  evl1muld  18699  frlmvscaval  19096  frlmsslsp  19123  frlmup1  19125  frlmup2  19126  islindf4  19165  mamudi  19197  mamudir  19198  mdetrlin  19396  nmotri  21538  mdegaddle  22766  ply1rem  22856  fta1glem2  22859  fta1blem  22861  plyexmo  23001  ulmdvlem1  23087  jensen  23644  dchrmulcl  23905  dchrinv  23917  sumdchr2  23926  dchr2sum  23929  ofccat  29003  mzpsubst  35042  mzpcong  35271  rngunsnply  35486  lincsum  38541
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