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Theorem istps2 21471
Description: Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Base‘𝐾)
istps.j 𝐽 = (TopOpen‘𝐾)
Assertion
Ref Expression
istps2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))

Proof of Theorem istps2
StepHypRef Expression
1 istps.a . . 3 𝐴 = (Base‘𝐾)
2 istps.j . . 3 𝐽 = (TopOpen‘𝐾)
31, 2istps 21470 . 2 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
4 istopon 21448 . 2 (𝐽 ∈ (TopOn‘𝐴) ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
53, 4bitri 276 1 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105   cuni 4830  cfv 6348  Basecbs 16471  TopOpenctopn 16683  Topctop 21429  TopOnctopon 21446  TopSpctps 21468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-top 21430  df-topon 21447  df-topsp 21469
This theorem is referenced by:  tpsuni  21472  tpstop  21473  istpsi  21478
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