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Theorem hmeocnvb 20565
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 20553 . . 3  |-  ( `' F  e.  ( J
Homeo K )  ->  `' `' F  e.  ( K Homeo J ) )
2 dfrel2 5273 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
3 eleq1 2474 . . . 4  |-  ( `' `' F  =  F  ->  ( `' `' F  e.  ( K Homeo J )  <-> 
F  e.  ( K
Homeo J ) ) )
42, 3sylbi 195 . . 3  |-  ( Rel 
F  ->  ( `' `' F  e.  ( K Homeo J )  <->  F  e.  ( K Homeo J ) ) )
51, 4syl5ib 219 . 2  |-  ( Rel 
F  ->  ( `' F  e.  ( J Homeo K )  ->  F  e.  ( K Homeo J ) ) )
6 hmeocnv 20553 . 2  |-  ( F  e.  ( K Homeo J )  ->  `' F  e.  ( J Homeo K ) )
75, 6impbid1 203 1  |-  ( Rel 
F  ->  ( `' F  e.  ( J Homeo K )  <->  F  e.  ( K Homeo J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   `'ccnv 4821   Rel wrel 4827  (class class class)co 6277   Homeochmeo 20544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-top 19689  df-topon 19692  df-cn 20019  df-hmeo 20546
This theorem is referenced by: (None)
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