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Theorem istpsi 22948
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 2746 . . 3 𝐴 = (Base‘𝐾)
5 istpsi.j . . . 4 (TopOpen‘𝐾) = 𝐽
65eqcomi 2746 . . 3 𝐽 = (TopOpen‘𝐾)
74, 6istps2 22941 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
81, 2, 7mpbir2an 711 1 𝐾 ∈ TopSp
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108   cuni 4907  cfv 6561  Basecbs 17247  TopOpenctopn 17466  Topctop 22899  TopSpctps 22938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-topon 22917  df-topsp 22939
This theorem is referenced by:  indistps2  23019
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