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Theorem fvco 5850
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 5525 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
2 fvco2 5849 . 2  |-  ( ( G  Fn  dom  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `
 A ) ) )
31, 2sylanb 470 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 367    = wceq 1399    e. wcel 1826   dom cdm 4913    o. ccom 4917   Fun wfun 5490    Fn wfn 5491   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504
This theorem is referenced by:  fin23lem30  8635  hashkf  12309  hashgval  12310  gsumpropd2lem  16017  ofco2  19038  opfv  27626  xppreima  27627  eulerpartlemgvv  28498  eulerpartlemgu  28499  sseqfv2  28516  evthiccabs  31695  cncficcgt0  31857  dvsinax  31874  stirlinglem14  32035  fourierdlem42  32097  comptiunov2i  38234
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