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Theorem tpstop 21153
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 2778 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 tpstop.j . . 3 𝐽 = (TopOpen‘𝐾)
31, 2istps2 21151 . 2 (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ (Base‘𝐾) = 𝐽))
43simplbi 493 1 (𝐾 ∈ TopSp → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107   cuni 4673  cfv 6137  Basecbs 16259  TopOpenctopn 16472  Topctop 21109  TopSpctps 21148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-top 21110  df-topon 21127  df-topsp 21149
This theorem is referenced by:  mreclatdemoBAD  21312  prdstmdd  22339  invrcn  22396  cnextucn  22519  prdsxmslem2  22746  rlmbn  23571  sibfinima  31003  sibfof  31004  rrxtop  41443
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