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Theorem tpspropd 22945
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 6909 . . 3 (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
41, 3eleq12d 2834 . 2 (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5 eqid 2736 . . 3 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2736 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
75, 6istps 22941 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8 eqid 2736 . . 3 (Base‘𝐿) = (Base‘𝐿)
9 eqid 2736 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
108, 9istps 22941 . 2 (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
114, 7, 103bitr4g 314 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1539  wcel 2107  cfv 6560  Basecbs 17248  TopOpenctopn 17467  TopOnctopon 22917  TopSpctps 22939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-topon 22918  df-topsp 22940
This theorem is referenced by:  tpsprop2d  22946  xmspropd  24484
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