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Theorem hmphtop 21391
 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 21369 . . 3 ≃ = (Homeo “ (V ∖ 1𝑜))
2 cnvimass 5404 . . . 4 (Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo
3 hmeofn 21370 . . . . 5 Homeo Fn (Top × Top)
4 fndm 5904 . . . . 5 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . 4 dom Homeo = (Top × Top)
62, 5sseqtri 3600 . . 3 (Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top)
71, 6eqsstri 3598 . 2 ≃ ⊆ (Top × Top)
87brel 5090 1 (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   class class class wbr 4583   × cxp 5036  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  1𝑜c1o 7440  Topctop 20517  Homeochmeo 21366   ≃ chmph 21367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-hmeo 21368  df-hmph 21369 This theorem is referenced by:  hmphtop1  21392  hmphtop2  21393
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