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Theorem f1f1orn 5346
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 5300 . 2  |-  ( F : A -1-1-> B  ->  F  Fn  A )
2 df-f1 5098 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
32simprbi 273 . 2  |-  ( F : A -1-1-> B  ->  Fun  `' F )
4 f1orn 5345 . 2  |-  ( F : A -1-1-onto-> ran  F  <->  ( F  Fn  A  /\  Fun  `' F ) )
51, 3, 4sylanbrc 413 1  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4   `'ccnv 4508   ran crn 4510   Fun wfun 5087    Fn wfn 5088   -->wf 5089   -1-1->wf1 5090   -1-1-onto->wf1o 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-in 3047  df-ss 3054  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1ores  5350  f1cnv  5359  f1cocnv1  5365  f1ocnvfvrneq  5651  ssenen  6713  f1dmvrnfibi  6800
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