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Theorem f1ocnvb 5647
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 5646 . 2  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1ocnv 5646 . . 3  |-  ( `' F : B -1-1-onto-> A  ->  `' `' F : A -1-1-onto-> B )
3 dfrel2 5280 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
4 f1oeq1 5624 . . . 4  |-  ( `' `' F  =  F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A -1-1-onto-> B ) )
53, 4sylbi 188 . . 3  |-  ( Rel 
F  ->  ( `' `' F : A -1-1-onto-> B  <->  F : A
-1-1-onto-> B ) )
62, 5syl5ib 211 . 2  |-  ( Rel 
F  ->  ( `' F : B -1-1-onto-> A  ->  F : A
-1-1-onto-> B ) )
71, 6impbid2 196 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 177    = wceq 1649   `'ccnv 4836   Rel wrel 4842   -1-1-onto->wf1o 5412
This theorem is referenced by:  hasheqf1oi  11590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
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