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Theorem hmph 17761
Description: Express the predicate  J is homeomorph to  K. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmph  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )

Proof of Theorem hmph
StepHypRef Expression
1 df-hmph 17741 . 2  |-  ~=  =  ( `'  Homeo  " ( _V  \  1o ) )
2 hmeofn 17742 . 2  |-  Homeo  Fn  ( Top  X.  Top )
31, 2brwitnlem 6710 1  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    =/= wne 2567   (/)c0 3588   class class class wbr 4172    X. cxp 4835  (class class class)co 6040   Topctop 16913    Homeo chmeo 17738    ~= chmph 17739
This theorem is referenced by:  hmphi  17762  hmphsym  17767  hmphtr  17768  hmphen  17770  haushmphlem  17772  cmphmph  17773  conhmph  17774  reghmph  17778  nrmhmph  17779  hmphdis  17781  hmphen2  17784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-hmeo 17740  df-hmph 17741
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