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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 6674 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿))) |
4 | 1, 3 | eleq12d 2907 | . 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))) |
5 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
7 | 5, 6 | istps 21542 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
8 | eqid 2821 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | eqid 2821 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
10 | 8, 9 | istps 21542 | . 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))) |
11 | 4, 7, 10 | 3bitr4g 316 | 1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 Basecbs 16483 TopOpenctopn 16695 TopOnctopon 21518 TopSpctps 21540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-top 21502 df-topon 21519 df-topsp 21541 |
This theorem is referenced by: tpsprop2d 21547 xmspropd 23083 |
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