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Theorem fnfco 5580
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 5425 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5522 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1188 . 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 3331   ran crn 4844    o. ccom 4847    Fn wfn 5416   -->wf 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pr 4534
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-br 4296  df-opab 4354  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-fun 5423  df-fn 5424  df-f 5425
This theorem is referenced by:  cocan1  5998  cocan2  5999  ofco  6343  1stcof  6607  2ndcof  6608  axcc3  8610  dmaf  14920  cdaf  14921  gsumzaddlem  16411  gsumzaddlemOLD  16413  prdstopn  19204  xpstopnlem2  19387  prdstgpd  19698  prdsxmslem2  20107  uniiccdif  21061  uniiccvol  21063  uniioombllem2  21066  resinf1o  21995  jensen  22385  occllem  24709  nlelchi  25468  hmopidmchi  25558  iprodefisumlem  27507  stoweidlem27  29825
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