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Theorem tpstop 20962
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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 tpstop.j . . 3 𝐽 = (TopOpen‘𝐾)
31, 2istps2 20960 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = 𝐽))
43simplbi 485 1 (𝐾 ∈ TopSp → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145   cuni 4574  cfv 6031  Basecbs 16064  TopOpenctopn 16290  Topctop 20918  TopSpctps 20957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-top 20919  df-topon 20936  df-topsp 20958
This theorem is referenced by:  mreclatdemoBAD  21121  prdstmdd  22147  invrcn  22204  cnextucn  22327  prdsxmslem2  22554  rlmbn  23376  sibfinima  30741  sibfof  30742  rrxtop  41026
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