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Theorem dff1o5 5344
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o5  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )

Proof of Theorem dff1o5
StepHypRef Expression
1 df-f1o 5100 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 f1f 5298 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
32biantrurd 303 . . . 4  |-  ( F : A -1-1-> B  -> 
( ran  F  =  B 
<->  ( F : A --> B  /\  ran  F  =  B ) ) )
4 dffo2 5319 . . . 4  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
53, 4syl6rbbr 198 . . 3  |-  ( F : A -1-1-> B  -> 
( F : A -onto-> B 
<->  ran  F  =  B ) )
65pm5.32i 449 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
71, 6bitri 183 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316   ran crn 4510   -->wf 5089   -1-1->wf1 5090   -onto->wfo 5091   -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-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:  f1orescnv  5351  f1finf1o  6803  djuinr  6916  eninl  6950  eninr  6951  frec2uzf1od  10147  ennnfonelemex  11854  ennnfonelemen  11861  pwf1oexmid  13121
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