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Theorem f1ocnvb 5267
Description: A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
f1ocnvb  |-  ( Rel 
F  ->  ( F : A -1-1-onto-> B  <->  `' F : B -1-1-onto-> A ) )

Proof of Theorem f1ocnvb
StepHypRef Expression
1 f1ocnv 5266 . 2  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1ocnv 5266 . . 3  |-  ( `' F : B -1-1-onto-> A  ->  `' `' F : A -1-1-onto-> B )
3 dfrel2 4881 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
4 f1oeq1 5244 . . . 4  |-  ( `' `' F  =  F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A -1-1-onto-> B ) )
53, 4sylbi 119 . . 3  |-  ( Rel 
F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
62, 5syl5ib 152 . 2  |-  ( Rel 
F  ->  ( `' F : B -1-1-onto-> A  ->  F : A
-1-1-onto-> B ) )
71, 6impbid2 141 1  |-  ( Rel 
F  ->  ( F : A -1-1-onto-> B  <->  `' F : B -1-1-onto-> A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   `'ccnv 4437   Rel wrel 4443   -1-1-onto->wf1o 5014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022
This theorem is referenced by: (None)
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