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Theorem f1ocnvb 5765
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 5764 . 2  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
2 f1ocnv 5764 . . 3  |-  ( `' F : B -1-1-onto-> A  ->  `' `' F : A -1-1-onto-> B )
3 dfrel2 5399 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
4 f1oeq1 5743 . . . 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 1370   `'ccnv 4950   Rel wrel 4956   -1-1-onto->wf1o 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536
This theorem is referenced by:  hasheqf1oi  12243
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