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Mirrors > Home > MPE Home > Th. List > istpsi | Structured version Visualization version GIF version |
Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
Ref | Expression |
---|---|
istpsi.b | ⊢ (Base‘𝐾) = 𝐴 |
istpsi.j | ⊢ (TopOpen‘𝐾) = 𝐽 |
istpsi.1 | ⊢ 𝐴 = ∪ 𝐽 |
istpsi.2 | ⊢ 𝐽 ∈ Top |
Ref | Expression |
---|---|
istpsi | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istpsi.2 | . 2 ⊢ 𝐽 ∈ Top | |
2 | istpsi.1 | . 2 ⊢ 𝐴 = ∪ 𝐽 | |
3 | istpsi.b | . . . 4 ⊢ (Base‘𝐾) = 𝐴 | |
4 | 3 | eqcomi 2832 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
5 | istpsi.j | . . . 4 ⊢ (TopOpen‘𝐾) = 𝐽 | |
6 | 5 | eqcomi 2832 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) |
7 | 4, 6 | istps2 21545 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
8 | 1, 2, 7 | mpbir2an 709 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∪ cuni 4840 ‘cfv 6357 Basecbs 16485 TopOpenctopn 16697 Topctop 21503 TopSpctps 21542 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-top 21504 df-topon 21521 df-topsp 21543 |
This theorem is referenced by: indistps2 21622 |
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