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Theorem dffo2 5110
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 5106 . . 3  |-  ( F : A -onto-> B  ->  F : A --> B )
2 forn 5109 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
31, 2jca 290 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  ran  F  =  B ) )
4 ffn 5046 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
5 df-fo 4908 . . . 4  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
65biimpri 124 . . 3  |-  ( ( F  Fn  A  /\  ran  F  =  B )  ->  F : A -onto-> B )
74, 6sylan 267 . 2  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A -onto-> B )
83, 7impbii 117 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   ran crn 4346    Fn wfn 4897   -->wf 4898   -onto->wfo 4900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906  df-fo 4908
This theorem is referenced by:  foco  5116  dff1o5  5135  dffo3  5314  dffo4  5315  fo1stresm  5788  fo2ndresm  5789  fo2ndf  5848  1fv  8994
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