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Theorem fvco 5868
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 5547 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
2 fvco2 5867 . 2  |-  ( ( G  Fn  dom  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `
 A ) ) )
31, 2sylanb 472 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 369    = wceq 1370    e. wcel 1758   dom cdm 4940    o. ccom 4944   Fun wfun 5512    Fn wfn 5513   ` cfv 5518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-fv 5526
This theorem is referenced by:  fin23lem30  8614  hashkf  12208  hashgval  12209  gsumpropd2lem  15609  ofco2  18445  opfv  26099  xppreima  26100  eulerpartlemgvv  26895  eulerpartlemgu  26896  sseqfv2  26913  stirlinglem14  30022
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