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Theorem fnfvof 7639
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 766 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐹 Fn 𝐴)
2 simplr 768 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐺 Fn 𝐴)
3 simpr 486 . . 3 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) → 𝐴𝑉)
4 inidm 4183 . . 3 (𝐴𝐴) = 𝐴
5 eqidd 2738 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐹𝑋) = (𝐹𝑋))
6 eqidd 2738 . . 3 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → (𝐺𝑋) = (𝐺𝑋))
71, 2, 3, 3, 4, 5, 6ofval 7633 . 2 ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐴𝑉) ∧ 𝑋𝐴) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
87anasss 468 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑋𝐴)) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   Fn wfn 6496  cfv 6501  (class class class)co 7362  f cof 7620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622
This theorem is referenced by:  suppofssd  8139  ofccat  14861  ghmplusg  19631  lcomfsupp  20378  lmhmplusg  20521  frlmvplusgvalc  21189  frlmvscaval  21190  frlmsslsp  21218  frlmup1  21220  frlmup2  21221  islindf4  21260  evlslem3  21506  evlslem1  21508  coe1addfv  21652  evl1addd  21723  evl1subd  21724  evl1muld  21725  mamudi  21766  mamudir  21767  mdetrlin  21967  nmotri  24119  mdegaddle  25455  ply1rem  25544  fta1glem2  25547  fta1blem  25549  plyexmo  25689  ulmdvlem1  25775  jensen  26354  dchrmulcl  26613  dchrinv  26625  sumdchr2  26634  dchr2sum  26637  evlsaddval  40779  evlsmulval  40780  evladdval  40782  evlmulval  40783  mzpsubst  41100  mzpcong  41325  rngunsnply  41529  ofoafg  41699  ofoafo  41701  ofoaid1  41703  ofoaid2  41704  ofoaass  41705  ofoacom  41706  naddcnff  41707  naddcnffo  41709  naddcnfcom  41711  naddcnfid1  41712  naddcnfass  41714  lincsum  46584
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