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Theorem fvco 5597
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 5285 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
2 fvco2 5596 . 2  |-  ( ( G  Fn  dom  G  /\  A  e.  dom  G )  ->  ( ( F  o.  G ) `  A )  =  ( F `  ( G `
 A ) ) )
31, 2sylanb 458 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( F  o.  G ) `  A
)  =  ( F `
 ( G `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   dom cdm 4691    o. ccom 4695   Fun wfun 5251    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  fin23lem30  7970  hashkf  11341  hashgval  11342  opfv  23192  xppreima  23213  gsumpropd2lem  23381  stirlinglem14  27847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265
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