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

Theorem fvco 6762
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 6388 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
2 fvco2 6761 . 2 ((𝐺 Fn dom 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
31, 2sylanb 583 1 ((Fun 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  dom cdm 5558  ccom 5562  Fun wfun 6352   Fn wfn 6353  cfv 6358
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366
This theorem is referenced by:  fin23lem30  9767  hashkf  13695  hashgval  13696  gsumpropd2lem  17892  ofco2  21063  opfv  30396  xppreima  30397  psgnfzto1stlem  30746  cycpmfv1  30759  cycpmfv2  30760  cyc3co2  30786  smatlem  31066  mdetpmtr1  31092  madjusmdetlem2  31097  madjusmdetlem4  31099  eulerpartlemgvv  31638  eulerpartlemgu  31639  sseqfv2  31656  reprpmtf1o  31901  hgt750lemg  31929  comptiunov2i  40057  choicefi  41469  fvcod  41498  evthiccabs  41777  cncficcgt0  42177  dvsinax  42203  fvvolioof  42281  fvvolicof  42283  stirlinglem14  42379  fourierdlem42  42441  hoicvr  42837  hoi2toco  42896  ovolval3  42936  ovolval4lem1  42938  ovnovollem1  42945  ovnovollem2  42946
  Copyright terms: Public domain W3C validator