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Theorem f1orn 5816
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 5811 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F ) )
2 eqid 2443 . . 3  |-  ran  F  =  ran  F
3 df-3an 976 . . 3  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  ran  F )  <->  ( ( F  Fn  A  /\  Fun  `' F )  /\  ran  F  =  ran  F ) )
42, 3mpbiran2 919 . 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 974    = wceq 1383   `'ccnv 4988   ran crn 4990   Fun wfun 5572    Fn wfn 5573   -1-1-onto->wf1o 5577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-in 3468  df-ss 3475  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585
This theorem is referenced by:  f1f1orn  5817  infdifsn  8076  efopnlem2  23014
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