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Theorem f1ocnvb 5649
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 5648 . 2  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1ocnv 5648 . . 3  |-  ( `' F : B -1-1-onto-> A  ->  `' `' F : A -1-1-onto-> B )
3 dfrel2 5283 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
4 f1oeq1 5627 . . . 4  |-  ( `' `' F  =  F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A -1-1-onto-> B ) )
53, 4sylbi 195 . . 3  |-  ( Rel 
F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
62, 5syl5ib 219 . 2  |-  ( Rel 
F  ->  ( `' F : B -1-1-onto-> A  ->  F : A
-1-1-onto-> B ) )
71, 6impbid2 204 1  |-  ( Rel 
F  ->  ( F : A -1-1-onto-> B  <->  `' F : B -1-1-onto-> A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369   `'ccnv 4834   Rel wrel 4840   -1-1-onto->wf1o 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
This theorem is referenced by:  hasheqf1oi  12114
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