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Theorem dff1o3 5620
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 5619 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
3 df-fo 5400 . . 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 1649   `'ccnv 4817   ran crn 4819   Fun wfun 5388    Fn wfn 5389   -onto->wfo 5392   -1-1-onto->wf1o 5393
This theorem is referenced by:  f1ofo  5621  resdif  5636  f11o  5648  f1opw  6238  1stconst  6374  2ndconst  6375  curry1  6377  curry2  6380  ssdomg  7089  phplem4  7225  php3  7229  f1opwfi  7345  cantnfp1lem3  7569  mapfien  7586  fpwwe2lem6  8443  canthp1lem2  8461  odf1o2  15134  dprdf1o  15517  relogf1o  20331  ballotlemfrc  24563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-in 3270  df-ss 3277  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401
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