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Theorem fvco 6437
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 6079 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 6436 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 490 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  dom cdm 5266  ccom 5270  Fun wfun 6043   Fn wfn 6044  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  fin23lem30  9376  hashkf  13333  hashgval  13334  gsumpropd2lem  17494  ofco2  20479  opfv  29778  xppreima  29779  psgnfzto1stlem  30180  smatlem  30193  mdetpmtr1  30219  madjusmdetlem2  30224  madjusmdetlem4  30226  eulerpartlemgvv  30768  eulerpartlemgu  30769  sseqfv2  30786  reprpmtf1o  31034  hgt750lemg  31062  comptiunov2i  38518  choicefi  39909  fvcod  39940  evthiccabs  40239  cncficcgt0  40622  dvsinax  40648  fvvolioof  40727  fvvolicof  40729  stirlinglem14  40825  fourierdlem42  40887  hoicvr  41286  hoi2toco  41345  ovolval3  41385  ovolval4lem1  41387  ovnovollem1  41394  ovnovollem2  41395
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