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Theorem fvco 6313
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 5956 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 6312 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 488 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  dom cdm 5143  ccom 5147  Fun wfun 5920   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  fin23lem30  9202  hashkf  13159  hashgval  13160  gsumpropd2lem  17320  ofco2  20305  opfv  29576  xppreima  29577  psgnfzto1stlem  29978  smatlem  29991  mdetpmtr1  30017  madjusmdetlem2  30022  madjusmdetlem4  30024  eulerpartlemgvv  30566  eulerpartlemgu  30567  sseqfv2  30584  reprpmtf1o  30832  hgt750lemg  30860  comptiunov2i  38315  choicefi  39706  fvcod  39737  evthiccabs  40036  cncficcgt0  40419  dvsinax  40445  fvvolioof  40524  fvvolicof  40526  stirlinglem14  40622  fourierdlem42  40684  hoicvr  41083  hoi2toco  41142  ovolval3  41182  ovolval4lem1  41184  ovnovollem1  41191  ovnovollem2  41192
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