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

Theorem hmphtop 23786
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmphtop (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))

Proof of Theorem hmphtop
StepHypRef Expression
1 df-hmph 23764 . . 3 ≃ = (Homeo “ (V ∖ 1o))
2 cnvimass 6100 . . . 4 (Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3 hmeofn 23765 . . . . 5 Homeo Fn (Top × Top)
4 fndm 6671 . . . . 5 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . 4 dom Homeo = (Top × Top)
62, 5sseqtri 4032 . . 3 (Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
71, 6eqsstri 4030 . 2 ≃ ⊆ (Top × Top)
87brel 5750 1 (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688   Fn wfn 6556  1oc1o 8499  Topctop 22899  Homeochmeo 23761  chmph 23762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-hmeo 23763  df-hmph 23764
This theorem is referenced by:  hmphtop1  23787  hmphtop2  23788
  Copyright terms: Public domain W3C validator