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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 6906 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿))) |
4 | 1, 3 | eleq12d 2823 | . 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))) |
5 | eqid 2728 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2728 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
7 | 5, 6 | istps 22864 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
8 | eqid 2728 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | eqid 2728 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
10 | 8, 9 | istps 22864 | . 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 1533 ∈ wcel 2098 ‘cfv 6553 Basecbs 17189 TopOpenctopn 17412 TopOnctopon 22840 TopSpctps 22862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-top 22824 df-topon 22841 df-topsp 22863 |
This theorem is referenced by: tpsprop2d 22869 xmspropd 24407 |
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