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Theorem fof 5137
Description: An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
Assertion
Ref Expression
fof  |-  ( F : A -onto-> B  ->  F : A --> B )

Proof of Theorem fof
StepHypRef Expression
1 eqimss 3052 . . 3  |-  ( ran 
F  =  B  ->  ran  F  C_  B )
21anim2i 334 . 2  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  ( F  Fn  A  /\  ran  F  C_  B ) )
3 df-fo 4938 . 2  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
4 df-f 4936 . 2  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
52, 3, 43imtr4i 199 1  |-  ( F : A -onto-> B  ->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    C_ wss 2974   ran crn 4372    Fn wfn 4927   -->wf 4928   -onto->wfo 4930
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-f 4936  df-fo 4938
This theorem is referenced by:  fofun  5138  fofn  5139  dffo2  5141  foima  5142  resdif  5179  ffoss  5189  fconstfvm  5411  cocan2  5459  foeqcnvco  5461  fornex  5773  algrflem  5881  algrflemg  5882  tposf2  5917  ssdomg  6325  fopwdom  6380  focdmex  9811
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