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Theorem hmphsym 17814
Description: "Is homeomorph to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.)
Assertion
Ref Expression
hmphsym  |-  ( J  ~=  K  ->  K  ~=  J )

Proof of Theorem hmphsym
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 hmph 17808 . . 3  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3637 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
31, 2bitri 241 . 2  |-  ( J  ~=  K  <->  E. f 
f  e.  ( J 
Homeo  K ) )
4 hmeocnv 17794 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  `' f  e.  ( K  Homeo  J ) )
5 hmphi 17809 . . . 4  |-  ( `' f  e.  ( K 
Homeo  J )  ->  K  ~=  J )
64, 5syl 16 . . 3  |-  ( f  e.  ( J  Homeo  K )  ->  K  ~=  J )
76exlimiv 1644 . 2  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  K  ~=  J )
83, 7sylbi 188 1  |-  ( J  ~=  K  ->  K  ~=  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    e. wcel 1725    =/= wne 2599   (/)c0 3628   class class class wbr 4212   `'ccnv 4877  (class class class)co 6081    Homeo chmeo 17785    ~= chmph 17786
This theorem is referenced by:  hmpher  17816  hmphsymb  17818  haushmphlem  17819  t0kq  17850  kqhmph  17851  ist1-5lem  17852  reheibor  26548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-1o 6724  df-map 7020  df-top 16963  df-topon 16966  df-cn 17291  df-hmeo 17787  df-hmph 17788
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