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Theorem dff1o3 5157
Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
dff1o3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )

Proof of Theorem dff1o3
StepHypRef Expression
1 3anan32 931 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F
) )
2 dff1o2 5156 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
3 df-fo 4932 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
43anbi1i 446 . 2  |-  ( ( F : A -onto-> B  /\  Fun  `' F )  <-> 
( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F ) )
51, 2, 43bitr4i 210 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   `'ccnv 4364   ran crn 4366   Fun wfun 4920    Fn wfn 4921   -onto->wfo 4924   -1-1-onto->wf1o 4925
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 4930  df-f1 4931  df-fo 4932  df-f1o 4933
This theorem is referenced by:  f1ofo  5158  resdif  5173  f11o  5184  f1opw  5732  1stconst  5867  2ndconst  5868  f1o2ndf1  5874  ssdomg  6317  phplem4  6380  phplem4on  6392
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