ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff1o4 Unicode version

Theorem dff1o4 5470
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 5467 . 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 4638 . . . . . 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 5220 . . . 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 4626   dom cdm 4627   ran crn 4628   Fun wfun 5211    Fn wfn 5212   -1-1-onto->wf1o 5216
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 3136  df-ss 3143  df-rn 4638  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224
This theorem is referenced by:  f1ocnv  5475  f1oun  5482  f1o00  5497  f1oi  5500  f1osn  5502  f1ompt  5668  f1ofveu  5863  f1ocnvd  6073  f1od2  6236  mapsnf1o2  6696  sbthlemi9  6964  mhmf1o  12861  grpinvf1o  12940  hmeof1o2  13811
  Copyright terms: Public domain W3C validator