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Theorem fvco 6169
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 5819 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 6168 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 487 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  dom cdm 5028  ccom 5032  Fun wfun 5784   Fn wfn 5785  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798
This theorem is referenced by:  fin23lem30  9024  hashkf  12936  hashgval  12937  gsumpropd2lem  17042  ofco2  20018  opfv  28634  xppreima  28635  psgnfzto1stlem  28987  smatlem  28997  mdetpmtr1  29023  madjusmdetlem2  29028  madjusmdetlem4  29030  eulerpartlemgvv  29571  eulerpartlemgu  29572  sseqfv2  29589  comptiunov2i  36820  choicefi  38190  fvcod  38221  evthiccabs  38369  cncficcgt0  38578  dvsinax  38605  fvvolioof  38686  fvvolicof  38688  stirlinglem14  38784  fourierdlem42  38846  hoicvr  39242  hoi2toco  39301  ovolval3  39341  ovolval4lem1  39343  ovnovollem1  39350  ovnovollem2  39351
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