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Theorem tpspropd 22271
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 6843 . . 3 (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
41, 3eleq12d 2832 . 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 22267 . 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 22267 . 2 (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
114, 7, 103bitr4g 313 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  cfv 6493  Basecbs 17075  TopOpenctopn 17295  TopOnctopon 22243  TopSpctps 22265
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-top 22227  df-topon 22244  df-topsp 22266
This theorem is referenced by:  tpsprop2d  22272  xmspropd  23810
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