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Theorem istps2 22300
Description: Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Baseβ€˜πΎ)
istps.j 𝐽 = (TopOpenβ€˜πΎ)
Assertion
Ref Expression
istps2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = βˆͺ 𝐽))

Proof of Theorem istps2
StepHypRef Expression
1 istps.a . . 3 𝐴 = (Baseβ€˜πΎ)
2 istps.j . . 3 𝐽 = (TopOpenβ€˜πΎ)
31, 2istps 22299 . 2 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜π΄))
4 istopon 22277 . 2 (𝐽 ∈ (TopOnβ€˜π΄) ↔ (𝐽 ∈ Top ∧ 𝐴 = βˆͺ 𝐽))
53, 4bitri 275 1 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = βˆͺ 𝐽))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4866  β€˜cfv 6497  Basecbs 17088  TopOpenctopn 17308  Topctop 22258  TopOnctopon 22275  TopSpctps 22297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-top 22259  df-topon 22276  df-topsp 22298
This theorem is referenced by:  tpsuni  22301  tpstop  22302  istpsi  22307
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