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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 6911 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿))) |
4 | 1, 3 | eleq12d 2833 | . 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))) |
5 | eqid 2735 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2735 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
7 | 5, 6 | istps 22956 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
8 | eqid 2735 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | eqid 2735 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
10 | 8, 9 | istps 22956 | . 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))) |
11 | 4, 7, 10 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 Basecbs 17245 TopOpenctopn 17468 TopOnctopon 22932 TopSpctps 22954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-top 22916 df-topon 22933 df-topsp 22955 |
This theorem is referenced by: tpsprop2d 22961 xmspropd 24499 |
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