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Theorem hmeocnv 20708
Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeocnv  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K Homeo J ) )

Proof of Theorem hmeocnv
StepHypRef Expression
1 hmeocnvcn 20707 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeocn 20706 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
3 eqid 2429 . . . . . 6  |-  U. J  =  U. J
4 eqid 2429 . . . . . 6  |-  U. K  =  U. K
53, 4cnf 20193 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
6 frel 5749 . . . . 5  |-  ( F : U. J --> U. K  ->  Rel  F )
72, 5, 63syl 18 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  Rel  F )
8 dfrel2 5306 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
97, 8sylib 199 . . 3  |-  ( F  e.  ( J Homeo K )  ->  `' `' F  =  F )
109, 2eqeltrd 2517 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' `' F  e.  ( J  Cn  K ) )
11 ishmeo 20705 . 2  |-  ( `' F  e.  ( K
Homeo J )  <->  ( `' F  e.  ( K  Cn  J )  /\  `' `' F  e.  ( J  Cn  K ) ) )
121, 10, 11sylanbrc 668 1  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K Homeo J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   U.cuni 4222   `'ccnv 4853   Rel wrel 4859   -->wf 5597  (class class class)co 6305    Cn ccn 20171   Homeochmeo 20699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-top 19852  df-topon 19854  df-cn 20174  df-hmeo 20701
This theorem is referenced by:  hmeocnvb  20720  hmphsym  20728  xpstopnlem2  20757
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