MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeocnv Structured version   Unicode version

Theorem hmeocnv 20026
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 20025 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeocn 20024 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
3 eqid 2467 . . . . . 6  |-  U. J  =  U. J
4 eqid 2467 . . . . . 6  |-  U. K  =  U. K
53, 4cnf 19541 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
6 frel 5734 . . . . 5  |-  ( F : U. J --> U. K  ->  Rel  F )
72, 5, 63syl 20 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  Rel  F )
8 dfrel2 5457 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
97, 8sylib 196 . . 3  |-  ( F  e.  ( J Homeo K )  ->  `' `' F  =  F )
109, 2eqeltrd 2555 . 2  |-  ( F  e.  ( J Homeo K )  ->  `' `' F  e.  ( J  Cn  K ) )
11 ishmeo 20023 . 2  |-  ( `' F  e.  ( K
Homeo J )  <->  ( `' F  e.  ( K  Cn  J )  /\  `' `' F  e.  ( J  Cn  K ) ) )
121, 10, 11sylanbrc 664 1  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K Homeo J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   U.cuni 4245   `'ccnv 4998   Rel wrel 5004   -->wf 5584  (class class class)co 6284    Cn ccn 19519   Homeochmeo 20017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-top 19194  df-topon 19197  df-cn 19522  df-hmeo 20019
This theorem is referenced by:  hmeocnvb  20038  hmphsym  20046  xpstopnlem2  20075
  Copyright terms: Public domain W3C validator