MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmph Structured version   Visualization version   GIF version

Theorem hmph 21560
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 21540 . 2 ≃ = (Homeo “ (V ∖ 1𝑜))
2 hmeofn 21541 . 2 Homeo Fn (Top × Top)
31, 2brwitnlem 7572 1 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wne 2791  c0 3907   class class class wbr 4644   × cxp 5102  (class class class)co 6635  Topctop 20679  Homeochmeo 21537  chmph 21538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-1o 7545  df-hmeo 21539  df-hmph 21540
This theorem is referenced by:  hmphi  21561  hmphsym  21566  hmphtr  21567  hmphen  21569  haushmphlem  21571  cmphmph  21572  connhmph  21573  reghmph  21577  nrmhmph  21578  hmphdis  21580  hmphen2  21583
  Copyright terms: Public domain W3C validator