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Theorem f1orn 5762
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 5757 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F ) )
2 eqid 2454 . . 3  |-  ran  F  =  ran  F
3 df-3an 967 . . 3  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( ( F  Fn  A  /\  Fun  `' F )  /\  ran  F  =  ran  F ) )
42, 3mpbiran2 910 . 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 965    = wceq 1370   `'ccnv 4950   ran crn 4952   Fun wfun 5523    Fn wfn 5524   -1-1-onto->wf1o 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3446  df-ss 3453  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536
This theorem is referenced by:  f1f1orn  5763  infdifsn  7977  efopnlem2  22245
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