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Theorem tpsprop2d 22946
Description: A topological space depends only on the base and topology components. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tpsprop2d.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
tpsprop2d.2 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
Assertion
Ref Expression
tpsprop2d (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Proof of Theorem tpsprop2d
StepHypRef Expression
1 tpsprop2d.1 . 2 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
2 tpsprop2d.2 . . 3 (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
31, 2topnpropd 17482 . 2 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
41, 3tpspropd 22945 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  TopSetcts 17304  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-rep 5278  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-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  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-iun 4992  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-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-rest 17468  df-topn 17469  df-top 22901  df-topon 22918  df-topsp 22940
This theorem is referenced by: (None)
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