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Theorem hmphi 17574
Description: If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphi  |-  ( F  e.  ( J  Homeo  K )  ->  J  ~=  K )

Proof of Theorem hmphi
StepHypRef Expression
1 ne0i 3537 . 2  |-  ( F  e.  ( J  Homeo  K )  ->  ( J  Homeo  K )  =/=  (/) )
2 hmph 17573 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
31, 2sylibr 203 1  |-  ( F  e.  ( J  Homeo  K )  ->  J  ~=  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1710    =/= wne 2521   (/)c0 3531   class class class wbr 4104  (class class class)co 5945    Homeo chmeo 17550    ~= chmph 17551
This theorem is referenced by:  hmphref  17578  hmphsym  17579  hmphtr  17580  indishmph  17595  ptcmpfi  17610  t0kq  17615  kqhmph  17616  xrhmph  18549  xrge0hmph  23474  reheibor  25886
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-suc 4480  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-1o 6566  df-hmeo 17552  df-hmph 17553
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