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Theorem istpsi 22788
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 2733 . . 3 𝐴 = (Base‘𝐾)
5 istpsi.j . . . 4 (TopOpen‘𝐾) = 𝐽
65eqcomi 2733 . . 3 𝐽 = (TopOpen‘𝐾)
74, 6istps2 22781 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
81, 2, 7mpbir2an 708 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098   cuni 4900  cfv 6534  Basecbs 17149  TopOpenctopn 17372  Topctop 22739  TopSpctps 22778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-top 22740  df-topon 22757  df-topsp 22779
This theorem is referenced by:  indistps2  22859
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