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Theorem dff1o4 5674
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 5671 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
2 3anass 940 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) ) )
3 df-rn 4881 . . . . . 6  |-  ran  F  =  dom  `' F
43eqeq1i 2442 . . . . 5  |-  ( ran 
F  =  B  <->  dom  `' F  =  B )
54anbi2i 676 . . . 4  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <-> 
( Fun  `' F  /\  dom  `' F  =  B ) )
6 df-fn 5449 . . . 4  |-  ( `' F  Fn  B  <->  ( Fun  `' F  /\  dom  `' F  =  B )
)
75, 6bitr4i 244 . . 3  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <->  `' F  Fn  B
)
87anbi2i 676 . 2  |-  ( ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
91, 2, 83bitri 263 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652   `'ccnv 4869   dom cdm 4870   ran crn 4871   Fun wfun 5440    Fn wfn 5441   -1-1-onto->wf1o 5445
This theorem is referenced by:  f1ocnv  5679  f1oun  5686  f1o00  5702  f1oi  5705  f1osn  5707  f1oprswap  5709  f1ompt  5883  f1ocnvd  6285  curry1  6430  curry2  6433  f1ofveu  6576  mapsnf1o2  7053  omxpenlem  7201  sbthlem9  7217  compssiso  8246  fsumrev  12554  fsumshft  12555  invf1o  13986  grpinvf1o  14853  ghmf1o  15027  srngf1o  15934  lmhmf1o  16114  hmeof1o2  17787  f1o3d  24033  fprodshft  25292  fprodrev  25293  axcontlem2  25896  cdleme51finvN  31290
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-rn 4881  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453
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