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Theorem fnfco 5096
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 4936 . 2  |-  ( G : B --> A  <->  ( G  Fn  B  /\  ran  G  C_  A ) )
2 fnco 5038 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
323expb 1140 . 2  |-  ( ( F  Fn  A  /\  ( G  Fn  B  /\  ran  G  C_  A
) )  ->  ( F  o.  G )  Fn  B )
41, 3sylan2b 281 1  |-  ( ( F  Fn  A  /\  G : B --> A )  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    C_ wss 2974   ran crn 4372    o. ccom 4375    Fn wfn 4927   -->wf 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936
This theorem is referenced by:  cocan1  5458  cocan2  5459  ofco  5760  1stcof  5821  2ndcof  5822
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