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Theorem hmph 23724
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 23704 . 2 ≃ = (Homeo “ (V ∖ 1o))
2 hmeofn 23705 . 2 Homeo Fn (Top × Top)
31, 2brwitnlem 8528 1 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wne 2929  c0 4322   class class class wbr 5149   × cxp 5676  (class class class)co 7419  Topctop 22839  Homeochmeo 23701  chmph 23702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-1o 8487  df-hmeo 23703  df-hmph 23704
This theorem is referenced by:  hmphi  23725  hmphsym  23730  hmphtr  23731  hmphen  23733  haushmphlem  23735  cmphmph  23736  connhmph  23737  reghmph  23741  nrmhmph  23742  hmphdis  23744  hmphen2  23747
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