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Theorem fvco 5969
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  G  /\  A  e.  dom  G )  -> 
( ( F  o.  G ) `  A
)  =  ( F `
 ( G `  A ) ) )

Proof of Theorem fvco
StepHypRef Expression
1 funfn 5634 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
2 fvco2 5968 . 2  |-  ( ( G  Fn  dom  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `
 A ) ) )
31, 2sylanb 479 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( F  o.  G ) `  A
)  =  ( F `
 ( G `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898   dom cdm 4856    o. ccom 4860   Fun wfun 5599    Fn wfn 5600   ` cfv 5605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-fv 5613
This theorem is referenced by:  fin23lem30  8803  hashkf  12555  hashgval  12556  gsumpropd2lem  16571  ofco2  19531  opfv  28300  xppreima  28301  psgnfzto1stlem  28664  smatlem  28674  mdetpmtr1  28700  madjusmdetlem2  28705  madjusmdetlem4  28707  eulerpartlemgvv  29259  eulerpartlemgu  29260  sseqfv2  29277  comptiunov2i  36344  choicefi  37535  evthiccabs  37678  cncficcgt0  37852  dvsinax  37869  fvvolioof  37953  fvvolicof  37955  stirlinglem14  38050  fourierdlem42  38113  fourierdlem42OLD  38114  hoicvr  38477  hoi2toco  38536  ovolval3  38576  ovolval4lem1  38578  ovnovollem1  38585  ovnovollem2  38586
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