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Theorem tpspropd 21549
Description: A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tpspropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
tpspropd.2 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Assertion
Ref Expression
tpspropd (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Proof of Theorem tpspropd
StepHypRef Expression
1 tpspropd.2 . . 3 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
2 tpspropd.1 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
32fveq2d 6665 . . 3 (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
41, 3eleq12d 2910 . 2 (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5 eqid 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2824 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
75, 6istps 21545 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8 eqid 2824 . . 3 (Base‘𝐿) = (Base‘𝐿)
9 eqid 2824 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
108, 9istps 21545 . 2 (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
114, 7, 103bitr4g 317 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  cfv 6343  Basecbs 16483  TopOpenctopn 16695  TopOnctopon 21521  TopSpctps 21543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-top 21505  df-topon 21522  df-topsp 21544
This theorem is referenced by:  tpsprop2d  21550  xmspropd  23086
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