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Theorem istpsi 20559
 Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istpsi.b (Base‘𝐾) = 𝐴
istpsi.j (TopOpen‘𝐾) = 𝐽
istpsi.1 𝐴 = 𝐽
istpsi.2 𝐽 ∈ Top
Assertion
Ref Expression
istpsi 𝐾 ∈ TopSp

Proof of Theorem istpsi
StepHypRef Expression
1 istpsi.2 . 2 𝐽 ∈ Top
2 istpsi.1 . 2 𝐴 = 𝐽
3 istpsi.b . . . 4 (Base‘𝐾) = 𝐴
43eqcomi 2619 . . 3 𝐴 = (Base‘𝐾)
5 istpsi.j . . . 4 (TopOpen‘𝐾) = 𝐽
65eqcomi 2619 . . 3 𝐽 = (TopOpen‘𝐾)
74, 6istps2 20552 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
81, 2, 7mpbir2an 957 1 𝐾 ∈ TopSp
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  ‘cfv 5804  Basecbs 15695  TopOpenctopn 15905  Topctop 20517  TopSpctps 20519 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-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  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-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-topon 20523  df-topsp 20524 This theorem is referenced by:  indistps2  20626
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