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Theorem dff1o3 5253
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 935 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F
) )
2 dff1o2 5252 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
3 df-fo 5016 . . 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 924    = wceq 1289   `'ccnv 4435   ran crn 4437   Fun wfun 5004    Fn wfn 5005   -onto->wfo 5008   -1-1-onto->wf1o 5009
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017
This theorem is referenced by:  f1ofo  5254  resdif  5269  f11o  5280  f1opw  5843  1stconst  5978  2ndconst  5979  f1o2ndf1  5985  ssdomg  6485  phplem4  6561  phplem4on  6573
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