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Mirrors > Home > MPE Home > Th. List > tpspropd | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
tpspropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
tpspropd.2 | ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Ref | Expression |
---|---|
tpspropd | ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpspropd.2 | . . 3 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) | |
2 | tpspropd.1 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
3 | 2 | fveq2d 6843 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿))) |
4 | 1, 3 | eleq12d 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) | |
7 | 5, 6 | istps 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) | |
10 | 8, 9 | istps 22267 | . 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))) |
11 | 4, 7, 10 | 3bitr4g 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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