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Theorem tpsprop2d 21236
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 16544 . 2 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
41, 3tpspropd 21235 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1522  wcel 2081  cfv 6230  Basecbs 16317  TopSetcts 16405  TopSpctps 21229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-1st 7550  df-2nd 7551  df-rest 16530  df-topn 16531  df-top 21191  df-topon 21208  df-topsp 21230
This theorem is referenced by: (None)
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