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Theorem dff1o4 5341
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 5338 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
2 3anass 949 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) ) )
3 df-rn 4518 . . . . . 6  |-  ran  F  =  dom  `' F
43eqeq1i 2123 . . . . 5  |-  ( ran 
F  =  B  <->  dom  `' F  =  B )
54anbi2i 450 . . . 4  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <-> 
( Fun  `' F  /\  dom  `' F  =  B ) )
6 df-fn 5094 . . . 4  |-  ( `' F  Fn  B  <->  ( Fun  `' F  /\  dom  `' F  =  B )
)
75, 6bitr4i 186 . . 3  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <->  `' F  Fn  B
)
87anbi2i 450 . 2  |-  ( ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
91, 2, 83bitri 205 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    /\ w3a 945    = wceq 1314   `'ccnv 4506   dom cdm 4507   ran crn 4508   Fun wfun 5085    Fn wfn 5086   -1-1-onto->wf1o 5090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052  df-rn 4518  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098
This theorem is referenced by:  f1ocnv  5346  f1oun  5353  f1o00  5368  f1oi  5371  f1osn  5373  f1ompt  5537  f1ofveu  5728  f1ocnvd  5938  f1od2  6098  mapsnf1o2  6556  sbthlemi9  6819  hmeof1o2  12372
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