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Theorem fnfvof 7417
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 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))

Proof of Theorem fnfvof
StepHypRef Expression
1 simpll 765 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐹 Fn 𝐴)
2 simplr 767 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐺 Fn 𝐴)
3 simpr 487 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐴𝑉)
4 inidm 4194 . . 3 (𝐴𝐴) = 𝐴
5 eqidd 2822 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐹𝑋))
6 eqidd 2822 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
71, 2, 3, 3, 4, 5, 6ofval 7412 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
87anasss 469 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110   Fn wfn 6344  cfv 6349  (class class class)co 7150  f cof 7401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403
This theorem is referenced by:  suppofssd  7861  ofccat  14323  ghmplusg  18960  lcomfsupp  19668  lmhmplusg  19810  evlslem3  20287  evlslem1  20289  coe1addfv  20427  evl1addd  20498  evl1subd  20499  evl1muld  20500  frlmvplusgvalc  20905  frlmvscaval  20906  frlmsslsp  20934  frlmup1  20936  frlmup2  20937  islindf4  20976  mamudi  21006  mamudir  21007  mdetrlin  21205  nmotri  23342  mdegaddle  24662  ply1rem  24751  fta1glem2  24754  fta1blem  24756  plyexmo  24896  ulmdvlem1  24982  jensen  25560  dchrmulcl  25819  dchrinv  25831  sumdchr2  25840  dchr2sum  25843  mzpsubst  39338  mzpcong  39562  rngunsnply  39766  lincsum  44478
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