Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvco Structured version   Visualization version   GIF version

Theorem fvco 6184
 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 5833 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 6183 . 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 1475   ∈ wcel 1977  dom cdm 5038   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  fin23lem30  9047  hashkf  12981  hashgval  12982  gsumpropd2lem  17096  ofco2  20076  opfv  28828  xppreima  28829  psgnfzto1stlem  29181  smatlem  29191  mdetpmtr1  29217  madjusmdetlem2  29222  madjusmdetlem4  29224  eulerpartlemgvv  29765  eulerpartlemgu  29766  sseqfv2  29783  comptiunov2i  37017  choicefi  38387  fvcod  38418  evthiccabs  38565  cncficcgt0  38774  dvsinax  38801  fvvolioof  38882  fvvolicof  38884  stirlinglem14  38980  fourierdlem42  39042  hoicvr  39438  hoi2toco  39497  ovolval3  39537  ovolval4lem1  39539  ovnovollem1  39546  ovnovollem2  39547
 Copyright terms: Public domain W3C validator