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Theorem dff1o3 5075
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 : -1-1-onto->  F : -onto->  Fun  `' F

Proof of Theorem dff1o3
StepHypRef Expression
1 3anan32 895 . 2  F  Fn  Fun  `' F  ran  F  F  Fn  ran  F  Fun  `' F
2 dff1o2 5074 . 2  F : -1-1-onto->  F  Fn  Fun  `' F  ran  F
3 df-fo 4851 . . 3  F : -onto->  F  Fn  ran  F
43anbi1i 431 . 2  F : -onto->  Fun  `' F  F  Fn  ran  F  Fun  `' F
51, 2, 43bitr4i 201 1  F : -1-1-onto->  F : -onto->  Fun  `' F
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   w3a 884   wceq 1242   `'ccnv 4287   ran crn 4289   Fun wfun 4839    Fn wfn 4840   -onto->wfo 4843   -1-1-onto->wf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852
This theorem is referenced by:  f1ofo  5076  resdif  5091  f11o  5102  f1opw  5649  1stconst  5784  2ndconst  5785  f1o2ndf1  5791  ssdomg  6194
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