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Theorem hmeocnvb 12513
Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmeocnvb  |-  ( Rel 
F  ->  ( `' F  e.  ( J Homeo K )  <->  F  e.  ( K Homeo J ) ) )

Proof of Theorem hmeocnvb
StepHypRef Expression
1 hmeocnv 12502 . . 3  |-  ( `' F  e.  ( J
Homeo K )  ->  `' `' F  e.  ( K Homeo J ) )
2 dfrel2 4992 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
3 eleq1 2202 . . . 4  |-  ( `' `' F  =  F  ->  ( `' `' F  e.  ( K Homeo J )  <-> 
F  e.  ( K
Homeo J ) ) )
42, 3sylbi 120 . . 3  |-  ( Rel 
F  ->  ( `' `' F  e.  ( K Homeo J )  <->  F  e.  ( K Homeo J ) ) )
51, 4syl5ib 153 . 2  |-  ( Rel 
F  ->  ( `' F  e.  ( J Homeo K )  ->  F  e.  ( K Homeo J ) ) )
6 hmeocnv 12502 . 2  |-  ( F  e.  ( K Homeo J )  ->  `' F  e.  ( J Homeo K ) )
75, 6impbid1 141 1  |-  ( Rel 
F  ->  ( `' F  e.  ( J Homeo K )  <->  F  e.  ( K Homeo J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   `'ccnv 4541   Rel wrel 4547  (class class class)co 5777   Homeochmeo 12495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-map 6547  df-top 12191  df-topon 12204  df-cn 12383  df-hmeo 12496
This theorem is referenced by: (None)
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