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Theorem dff1o3 5672
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 948 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F
) )
2 dff1o2 5671 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
3 df-fo 5452 . . 3  |-  ( F : A -onto-> B  <->  ( F  Fn  A  /\  ran  F  =  B ) )
43anbi1i 677 . 2  |-  ( ( F : A -onto-> B  /\  Fun  `' F )  <-> 
( ( F  Fn  A  /\  ran  F  =  B )  /\  Fun  `' F ) )
51, 2, 43bitr4i 269 1  |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652   `'ccnv 4869   ran crn 4871   Fun wfun 5440    Fn wfn 5441   -onto->wfo 5444   -1-1-onto->wf1o 5445
This theorem is referenced by:  f1ofo  5673  resdif  5688  f11o  5700  f1opw  6291  1stconst  6427  2ndconst  6428  curry1  6430  curry2  6433  f1o2ndf1  6446  ssdomg  7145  phplem4  7281  php3  7285  f1opwfi  7402  cantnfp1lem3  7628  mapfien  7645  fpwwe2lem6  8502  canthp1lem2  8520  odf1o2  15199  dprdf1o  15582  relogf1o  20456  ballotlemfrc  24776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-in 3319  df-ss 3326  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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