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Theorem fnfco 5741
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnfco  |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B
)

Proof of Theorem fnfco
StepHypRef Expression
1 df-f 5583 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5680 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1192 . 2  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  ran  G  C_  A
) )  ->  ( F  o.  G )  Fn  B )
41, 3sylan2b 475 1  |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    C_ wss 3469   ran crn 4993    o. ccom 4996    Fn wfn 5574   -->wf 5575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-fun 5581  df-fn 5582  df-f 5583
This theorem is referenced by:  cocan1  6173  cocan2  6174  ofco  6535  1stcof  6802  2ndcof  6803  axcc3  8807  dmaf  15223  cdaf  15224  gsumzaddlem  16718  gsumzaddlemOLD  16720  prdstopn  19857  xpstopnlem2  20040  prdstgpd  20351  prdsxmslem2  20760  uniiccdif  21715  uniiccvol  21717  uniioombllem2  21720  resinf1o  22649  jensen  23039  occllem  25883  nlelchi  26642  hmopidmchi  26732  iprodefisumlem  28686  stoweidlem27  31282
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