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Theorem fnfco 5574
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 5419 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5516 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1183 . 2  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  ran  G  C_  A
) )  ->  ( F  o.  G )  Fn  B )
41, 3sylan2b 472 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 3325   ran crn 4837    o. ccom 4840    Fn wfn 5410   -->wf 5411
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 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-fun 5417  df-fn 5418  df-f 5419
This theorem is referenced by:  cocan1  5992  cocan2  5993  ofco  6339  1stcof  6603  2ndcof  6604  axcc3  8603  dmaf  14913  cdaf  14914  gsumzaddlem  16401  gsumzaddlemOLD  16403  prdstopn  19160  xpstopnlem2  19343  prdstgpd  19654  prdsxmslem2  20063  uniiccdif  21017  uniiccvol  21019  uniioombllem2  21022  resinf1o  21951  jensen  22341  occllem  24641  nlelchi  25400  hmopidmchi  25490  iprodefisumlem  27433  stoweidlem27  29747
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