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Theorem dff1o4 5468
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
dff1o4  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )

Proof of Theorem dff1o4
StepHypRef Expression
1 dff1o2 5465 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
2 3anass 982 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) ) )
3 df-rn 4636 . . . . . 6  |-  ran  F  =  dom  `' F
43eqeq1i 2185 . . . . 5  |-  ( ran 
F  =  B  <->  dom  `' F  =  B )
54anbi2i 457 . . . 4  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <-> 
( Fun  `' F  /\  dom  `' F  =  B ) )
6 df-fn 5218 . . . 4  |-  ( `' F  Fn  B  <->  ( Fun  `' F  /\  dom  `' F  =  B )
)
75, 6bitr4i 187 . . 3  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <->  `' F  Fn  B
)
87anbi2i 457 . 2  |-  ( ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
91, 2, 83bitri 206 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   `'ccnv 4624   dom cdm 4625   ran crn 4626   Fun wfun 5209    Fn wfn 5210   -1-1-onto->wf1o 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-rn 4636  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222
This theorem is referenced by:  f1ocnv  5473  f1oun  5480  f1o00  5495  f1oi  5498  f1osn  5500  f1ompt  5666  f1ofveu  5860  f1ocnvd  6070  f1od2  6233  mapsnf1o2  6693  sbthlemi9  6961  mhmf1o  12793  grpinvf1o  12872  hmeof1o2  13679
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