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Theorem fnfco 5407
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 5259 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5352 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1152 . 2  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  ran  G  C_  A
) )  ->  ( F  o.  G )  Fn  B )
41, 3sylan2b 461 1  |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3152   ran crn 4690    o. ccom 4693    Fn wfn 5250   -->wf 5251
This theorem is referenced by:  cocan1  5801  cocan2  5802  ofco  6097  1stcof  6147  2ndcof  6148  axcc3  8064  dmaf  13881  cdaf  13882  gsumzaddlem  15203  prdstopn  17322  xpstopnlem2  17502  prdstgpd  17807  prdsxmslem2  18075  uniiccdif  18933  uniiccvol  18935  uniioombllem2  18938  resinf1o  19898  jensen  20283  occllem  21882  nlelchi  22641  hmopidmchi  22731  stoweidlem27  27776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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