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Theorem dffo2 5138
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 5134 . . 3  |-  ( F : A -onto-> B  ->  F : A --> B )
2 forn 5137 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
31, 2jca 294 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  ran  F  =  B ) )
4 ffn 5074 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
5 df-fo 4936 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
65biimpri 128 . . 3  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A -onto-> B )
74, 6sylan 271 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A -onto-> B )
83, 7impbii 121 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259   ran crn 4374    Fn wfn 4925   -->wf 4926   -onto->wfo 4928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-ss 2959  df-f 4934  df-fo 4936
This theorem is referenced by:  foco  5144  dff1o5  5163  dffo3  5342  dffo4  5343  fo1stresm  5816  fo2ndresm  5817  fo2ndf  5876  1fv  9098
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