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Theorem tpstop 22302
Description: The topology extractor on a topological space is a topology. (Contributed by FL, 27-Jun-2014.)
Hypothesis
Ref Expression
tpstop.j 𝐽 = (TopOpen‘𝐾)
Assertion
Ref Expression
tpstop (𝐾 ∈ TopSp → 𝐽 ∈ Top)

Proof of Theorem tpstop
StepHypRef Expression
1 eqid 2737 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 tpstop.j . . 3 𝐽 = (TopOpen‘𝐾)
31, 2istps2 22300 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = 𝐽))
43simplbi 499 1 (𝐾 ∈ TopSp → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   cuni 4870  cfv 6501  Basecbs 17090  TopOpenctopn 17310  Topctop 22258  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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-top 22259  df-topon 22276  df-topsp 22298
This theorem is referenced by:  mreclatdemoBAD  22463  prdstmdd  23491  invrcn  23548  cnextucn  23671  prdsxmslem2  23901  rlmbn  24741  sibfinima  32979  sibfof  32980  rrxtop  44604
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