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Theorem fvco 7007
Description: Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
Assertion
Ref Expression
fvco ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))

Proof of Theorem fvco
StepHypRef Expression
1 funfn 6596 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 7006 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 581 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  dom cdm 5685  ccom 5689  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  fin23lem30  10382  hashkf  14371  hashgval  14372  gsumpropd2lem  18692  ofco2  22457  opfv  32654  xppreima  32655  psgnfzto1stlem  33120  cycpmfv1  33133  cycpmfv2  33134  cyc3co2  33160  smatlem  33796  mdetpmtr1  33822  madjusmdetlem2  33827  madjusmdetlem4  33829  eulerpartlemgvv  34378  eulerpartlemgu  34379  sseqfv2  34396  reprpmtf1o  34641  hgt750lemg  34669  aks5lem2  42188  comptiunov2i  43719  choicefi  45205  fvcod  45232  evthiccabs  45509  cncficcgt0  45903  dvsinax  45928  fvvolioof  46004  fvvolicof  46006  stirlinglem14  46102  fourierdlem42  46164  hoicvr  46563  hoi2toco  46622  ovolval3  46662  ovolval4lem1  46664  ovnovollem1  46671  ovnovollem2  46672
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