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Theorem fnfvof 6809
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 786 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐹 Fn 𝐴)
2 simplr 788 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐺 Fn 𝐴)
3 simpr 476 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐴𝑉)
4 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
5 eqidd 2611 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐹𝑋))
6 eqidd 2611 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
71, 2, 3, 3, 4, 5, 6ofval 6804 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
87anasss 677 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹𝑓 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977   Fn wfn 5799  cfv 5804  (class class class)co 6549  𝑓 cof 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795
This theorem is referenced by:  ofccat  13556  ghmplusg  18072  lcomfsupp  18726  lmhmplusg  18865  evlslem3  19335  evlslem1  19336  coe1addfv  19456  evl1addd  19526  evl1subd  19527  evl1muld  19528  frlmvscaval  19929  frlmsslsp  19954  frlmup1  19956  frlmup2  19957  islindf4  19996  mamudi  20028  mamudir  20029  mdetrlin  20227  nmotri  22353  mdegaddle  23638  ply1rem  23727  fta1glem2  23730  fta1blem  23732  plyexmo  23872  ulmdvlem1  23958  jensen  24515  dchrmulcl  24774  dchrinv  24786  sumdchr2  24795  dchr2sum  24798  mzpsubst  36329  mzpcong  36557  rngunsnply  36762  lincsum  42012
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