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Theorem hmph 22370
 Description: Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmph (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)

Proof of Theorem hmph
StepHypRef Expression
1 df-hmph 22350 . 2 ≃ = (Homeo “ (V ∖ 1o))
2 hmeofn 22351 . 2 Homeo Fn (Top × Top)
31, 2brwitnlem 8115 1 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ≠ wne 3013  ∅c0 4274   class class class wbr 5047   × cxp 5534  (class class class)co 7138  Topctop 21487  Homeochmeo 22347   ≃ chmph 22348 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-1o 8085  df-hmeo 22349  df-hmph 22350 This theorem is referenced by:  hmphi  22371  hmphsym  22376  hmphtr  22377  hmphen  22379  haushmphlem  22381  cmphmph  22382  connhmph  22383  reghmph  22387  nrmhmph  22388  hmphdis  22390  hmphen2  22393
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