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Theorem fnfvof 6336
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 753 . . 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 461 . . 3  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  ->  A  e.  V )
4 inidm 3562 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2444 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2444 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6332 . 2  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  (
( F  oF R G ) `  X )  =  ( ( F `  X
) R ( G `
 X ) ) )
87anasss 647 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 369    = wceq 1369    e. wcel 1756    Fn wfn 5416   ` cfv 5421  (class class class)co 6094    oFcof 6321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323
This theorem is referenced by:  ghmplusg  16331  lcomfsupOLD  16987  lcomfsupp  16988  lmhmplusg  17128  evlslem3  17603  evlslem1  17604  coe1addfv  17722  evl1addd  17778  evl1subd  17779  evl1muld  17780  frlmvscaval  18197  frlmsslsp  18226  frlmsslspOLD  18227  frlmup1  18229  frlmup2  18230  islindf4  18270  mamudi  18310  mamudir  18311  mdetrlin  18412  nmotri  20321  mdegaddle  21548  ply1rem  21638  fta1glem2  21641  fta1blem  21643  plyexmo  21782  ulmdvlem1  21868  jensen  22385  dchrmulcl  22591  dchrinv  22603  sumdchr2  22612  dchr2sum  22615  ofccat  26944  mzpsubst  29088  mzpcong  29318  rngunsnply  29533  lincsum  30966
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