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Theorem f1orn 5498
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 5493 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F ) )
2 eqid 2296 . . 3  |-  ran  F  =  ran  F
3 df-3an 936 . . 3  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( ( F  Fn  A  /\  Fun  `' F )  /\  ran  F  =  ran  F ) )
42, 3mpbiran2 885 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( F  Fn  A  /\  Fun  `' F ) )
51, 4bitri 240 1  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632   `'ccnv 4704   ran crn 4706   Fun wfun 5265    Fn wfn 5266   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1f1orn  5499  infdifsn  7373  efopnlem2  20020
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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