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Theorem tpsprop2d 21690
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 16813 . 2 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
41, 3tpspropd 21689 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1542  wcel 2114  cfv 6339  Basecbs 16586  TopSetcts 16674  TopSpctps 21683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-rest 16799  df-topn 16800  df-top 21645  df-topon 21662  df-topsp 21684
This theorem is referenced by: (None)
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