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Theorem dff1o5 5748
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 5523 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
2 f1f 5704 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
32biantrurd 508 . . . 4  |-  ( F : A -1-1-> B  -> 
( ran  F  =  B 
<->  ( F : A --> B  /\  ran  F  =  B ) ) )
4 dffo2 5722 . . . 4  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
53, 4syl6rbbr 264 . . 3  |-  ( F : A -1-1-> B  -> 
( F : A -onto-> B 
<->  ran  F  =  B ) )
65pm5.32i 637 . 2  |-  ( ( F : A -1-1-> B  /\  F : A -onto-> B
)  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
71, 6bitri 249 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   ran crn 4939   -->wf 5512   -1-1->wf1 5513   -onto->wfo 5514   -1-1-onto->wf1o 5515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-in 3433  df-ss 3440  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523
This theorem is referenced by:  f1orescnv  5754  domdifsn  7494  sucdom2  7608  ackbij1  8508  ackbij2  8513  fin4en1  8579  om2uzf1oi  11877  s4f1o  12630  fvcosymgeq  16036  indlcim  18378  ausisusgra  23414  pwssplit4  29580  cdleme50f1o  34496  diaf1oN  35081
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