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Theorem hmphtop 21491
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 21469 . . 3 ≃ = (Homeo “ (V ∖ 1𝑜))
2 cnvimass 5444 . . . 4 (Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3 hmeofn 21470 . . . . 5 Homeo Fn (Top × Top)
4 fndm 5948 . . . . 5 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . 4 dom Homeo = (Top × Top)
62, 5sseqtri 3616 . . 3 (Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
71, 6eqsstri 3614 . 2 ≃ ⊆ (Top × Top)
87brel 5128 1 (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552   class class class wbr 4613   × cxp 5072  ccnv 5073  dom cdm 5074  cima 5077   Fn wfn 5842  1𝑜c1o 7498  Topctop 20617  Homeochmeo 21466  chmph 21467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-hmeo 21468  df-hmph 21469
This theorem is referenced by:  hmphtop1  21492  hmphtop2  21493
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