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Theorem f1f1orn 5168
Description: A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1f1orn  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )

Proof of Theorem f1f1orn
StepHypRef Expression
1 f1fn 5124 . 2  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 df-f1 4937 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 269 . 2  |-  ( F : A -1-1-> B  ->  Fun  `' F )
4 f1orn 5167 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F ) )
51, 3, 4sylanbrc 408 1  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4   `'ccnv 4370   ran crn 4372   Fun wfun 4926    Fn wfn 4927   -->wf 4928   -1-1->wf1 4929   -1-1-onto->wf1o 4931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939
This theorem is referenced by:  f1ores  5172  f1cnv  5181  f1cocnv1  5187  f1ocnvfvrneq  5453  f1dmvrnfibi  6452
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