ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffo2 Unicode version

Theorem dffo2 5349
Description: Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
dffo2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )

Proof of Theorem dffo2
StepHypRef Expression
1 fof 5345 . . 3  |-  ( F : A -onto-> B  ->  F : A --> B )
2 forn 5348 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
31, 2jca 304 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  ran  F  =  B ) )
4 ffn 5272 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
5 df-fo 5129 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
65biimpri 132 . . 3  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A -onto-> B )
74, 6sylan 281 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A -onto-> B )
83, 7impbii 125 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331   ran crn 4540    Fn wfn 5118   -->wf 5119   -onto->wfo 5121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127  df-fo 5129
This theorem is referenced by:  foco  5355  dff1o5  5376  dffo3  5567  dffo4  5568  fo1stresm  6059  fo2ndresm  6060  fo2ndf  6124  1fv  9928
  Copyright terms: Public domain W3C validator