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Theorem fnfco 5733
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 5573 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5670 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1198 . 2  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  ran  G  C_  A
) )  ->  ( F  o.  G )  Fn  B )
41, 3sylan2b 473 1  |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    C_ wss 3414   ran crn 4824    o. ccom 4827    Fn wfn 5564   -->wf 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-fun 5571  df-fn 5572  df-f 5573
This theorem is referenced by:  cocan1  6177  cocan2  6178  ofco  6542  1stcof  6812  2ndcof  6813  axcc3  8850  dmaf  15652  cdaf  15653  gsumzaddlem  17258  gsumzaddlemOLD  17260  prdstopn  20421  xpstopnlem2  20604  prdstgpd  20915  prdsxmslem2  21324  uniiccdif  22279  uniiccvol  22281  uniioombllem2  22284  resinf1o  23215  jensen  23644  occllem  26635  nlelchi  27393  hmopidmchi  27483  iprodefisumlem  29949  stoweidlem27  37177
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