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Theorem hmphsym 17489
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 17483 . . 3  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3477 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
31, 2bitri 240 . 2  |-  ( J  ~=  K  <->  E. f 
f  e.  ( J 
Homeo  K ) )
4 hmeocnv 17469 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  `' f  e.  ( K  Homeo  J ) )
5 hmphi 17484 . . . 4  |-  ( `' f  e.  ( K 
Homeo  J )  ->  K  ~=  J )
64, 5syl 15 . . 3  |-  ( f  e.  ( J  Homeo  K )  ->  K  ~=  J )
76exlimiv 1624 . 2  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  K  ~=  J )
83, 7sylbi 187 1  |-  ( J  ~=  K  ->  K  ~=  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    e. wcel 1696    =/= wne 2459   (/)c0 3468   class class class wbr 4039   `'ccnv 4704  (class class class)co 5874    Homeo chmeo 17460    ~= chmph 17461
This theorem is referenced by:  hmpher  17491  hmphsymb  17493  haushmphlem  17494  t0kq  17525  kqhmph  17526  ist1-5lem  17527  reheibor  26666
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462  df-hmph 17463
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