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Theorem f1orn 5831
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
f1orn  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F ) )

Proof of Theorem f1orn
StepHypRef Expression
1 dff1o2 5826 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F ) )
2 eqid 2467 . . 3  |-  ran  F  =  ran  F
3 df-3an 975 . . 3  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( ( F  Fn  A  /\  Fun  `' F )  /\  ran  F  =  ran  F ) )
42, 3mpbiran2 917 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( F  Fn  A  /\  Fun  `' F ) )
51, 4bitri 249 1  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   `'ccnv 5003   ran crn 5005   Fun wfun 5587    Fn wfn 5588   -1-1-onto->wf1o 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-in 3488  df-ss 3495  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600
This theorem is referenced by:  f1f1orn  5832  infdifsn  8083  efopnlem2  22881
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