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Theorem fnfvof 6864
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 789 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐹 Fn 𝐴)
2 simplr 791 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐺 Fn 𝐴)
3 simpr 477 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐴𝑉)
4 inidm 3800 . . 3 (𝐴𝐴) = 𝐴
5 eqidd 2622 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐹𝑋))
6 eqidd 2622 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
71, 2, 3, 3, 4, 5, 6ofval 6859 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
87anasss 678 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   Fn wfn 5842  cfv 5847  (class class class)co 6604  𝑓 cof 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850
This theorem is referenced by:  ofccat  13642  ghmplusg  18170  lcomfsupp  18824  lmhmplusg  18963  evlslem3  19433  evlslem1  19434  coe1addfv  19554  evl1addd  19624  evl1subd  19625  evl1muld  19626  frlmvscaval  20029  frlmsslsp  20054  frlmup1  20056  frlmup2  20057  islindf4  20096  mamudi  20128  mamudir  20129  mdetrlin  20327  nmotri  22453  mdegaddle  23738  ply1rem  23827  fta1glem2  23830  fta1blem  23832  plyexmo  23972  ulmdvlem1  24058  jensen  24615  dchrmulcl  24874  dchrinv  24886  sumdchr2  24895  dchr2sum  24898  mzpsubst  36788  mzpcong  37016  rngunsnply  37221  lincsum  41503
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