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Theorem hmphtop 23515
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 23493 . . 3 ≃ = (Homeo “ (V ∖ 1o))
2 cnvimass 6080 . . . 4 (Homeo “ (V ∖ 1o)) ⊆ dom Homeo
3 hmeofn 23494 . . . . 5 Homeo Fn (Top × Top)
4 fndm 6652 . . . . 5 (Homeo Fn (Top × Top) → dom Homeo = (Top × Top))
53, 4ax-mp 5 . . . 4 dom Homeo = (Top × Top)
62, 5sseqtri 4018 . . 3 (Homeo “ (V ∖ 1o)) ⊆ (Top × Top)
71, 6eqsstri 4016 . 2 ≃ ⊆ (Top × Top)
87brel 5741 1 (𝐽𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cdif 3945   class class class wbr 5148   × cxp 5674  ccnv 5675  dom cdm 5676  cima 5679   Fn wfn 6538  1oc1o 8465  Topctop 22628  Homeochmeo 23490  chmph 23491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-hmeo 23492  df-hmph 23493
This theorem is referenced by:  hmphtop1  23516  hmphtop2  23517
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