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Theorem fnfvof 6528
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 3700 . . 3  |-  ( A  i^i  A )  =  A
5 eqidd 2461 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( F `  X )  =  ( F `  X ) )
6 eqidd 2461 . . 3  |-  ( ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A  e.  V )  /\  X  e.  A )  ->  ( G `  X )  =  ( G `  X ) )
71, 2, 3, 3, 4, 5, 6ofval 6524 . 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 1374    e. wcel 1762    Fn wfn 5574   ` cfv 5579  (class class class)co 6275    oFcof 6513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515
This theorem is referenced by:  ghmplusg  16638  lcomfsupOLD  17325  lcomfsupp  17326  lmhmplusg  17466  evlslem3  17947  evlslem1  17948  coe1addfv  18070  evl1addd  18141  evl1subd  18142  evl1muld  18143  frlmvscaval  18560  frlmsslsp  18589  frlmsslspOLD  18590  frlmup1  18592  frlmup2  18593  islindf4  18633  mamudi  18665  mamudir  18666  mdetrlin  18864  nmotri  20974  mdegaddle  22202  ply1rem  22292  fta1glem2  22295  fta1blem  22297  plyexmo  22436  ulmdvlem1  22522  jensen  23039  dchrmulcl  23245  dchrinv  23257  sumdchr2  23266  dchr2sum  23269  ofccat  28123  mzpsubst  30272  mzpcong  30501  rngunsnply  30716  lincsum  31978
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