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Theorem dff1o3 5478
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 946 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F
) )
2 dff1o2 5477 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
3 df-fo 5261 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
43anbi1i 676 . 2  |-  ( ( F : A -onto-> B  /\  Fun  `' F )  <-> 
( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F ) )
51, 2, 43bitr4i 268 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623   `'ccnv 4688   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  f1ofo  5479  resdif  5494  f11o  5506  f1opw  6072  1stconst  6207  2ndconst  6208  curry1  6210  curry2  6213  ssdomg  6907  phplem4  7043  php3  7047  f1opwfi  7159  cantnfp1lem3  7382  mapfien  7399  fpwwe2lem6  8257  canthp1lem2  8275  odf1o2  14884  dprdf1o  15267  relogf1o  19924  ballotlemfrc  23085  domrancur1b  25200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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