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Theorem fvco 5447
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 5141 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
2 fvco2 5446 . 2  |-  ( ( G  Fn  dom  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `
 A ) ) )
31, 2sylanb 460 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( F  o.  G ) `  A
)  =  ( F `
 ( G `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   dom cdm 4580    o. ccom 4584   Fun wfun 4586    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  fin23lem30  7852  hashkf  11217  hashgval  11218
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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