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Theorem dff1o5 5186
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 4959 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 f1f 5143 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
32biantrurd 299 . . . 4  |-  ( F : A -1-1-> B  -> 
( ran  F  =  B 
<->  ( F : A --> B  /\  ran  F  =  B ) ) )
4 dffo2 5161 . . . 4  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
53, 4syl6rbbr 197 . . 3  |-  ( F : A -1-1-> B  -> 
( F : A -onto-> B 
<->  ran  F  =  B ) )
65pm5.32i 442 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
71, 6bitri 182 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1285   ran crn 4392   -->wf 4948   -1-1->wf1 4949   -onto->wfo 4950   -1-1-onto->wf1o 4951
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959
This theorem is referenced by:  f1orescnv  5193  f1finf1o  6486  frec2uzf1od  9540
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