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Theorem dff1o4 5480
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 5477 . 2  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
2 3anass 938 . 2  |-  ( ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B )  <->  ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) ) )
3 df-rn 4700 . . . . . 6  |-  ran  F  =  dom  `' F
43eqeq1i 2290 . . . . 5  |-  ( ran 
F  =  B  <->  dom  `' F  =  B )
54anbi2i 675 . . . 4  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <-> 
( Fun  `' F  /\  dom  `' F  =  B ) )
6 df-fn 5258 . . . 4  |-  ( `' F  Fn  B  <->  ( Fun  `' F  /\  dom  `' F  =  B )
)
75, 6bitr4i 243 . . 3  |-  ( ( Fun  `' F  /\  ran  F  =  B )  <->  `' F  Fn  B
)
87anbi2i 675 . 2  |-  ( ( F  Fn  A  /\  ( Fun  `' F  /\  ran  F  =  B ) )  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
91, 2, 83bitri 262 1  |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623   `'ccnv 4688   dom cdm 4689   ran crn 4690   Fun wfun 5249    Fn wfn 5250   -1-1-onto->wf1o 5254
This theorem is referenced by:  f1ocnv  5485  f1oun  5492  f1o00  5508  f1oi  5511  f1osn  5513  f1oprswap  5515  f1ompt  5682  f1ocnvd  6066  curry1  6210  curry2  6213  f1ofveu  6339  mapsnf1o2  6815  omxpenlem  6963  sbthlem9  6979  compssiso  8000  fsumrev  12241  fsumshft  12242  invf1o  13671  grpinvf1o  14538  ghmf1o  14712  srngf1o  15619  lmhmf1o  15803  hmeof1o2  17454  f1o3d  23037  axcontlem2  24593  trinv  25395  cdleme51finvN  30745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-rn 4700  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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