| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2746 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
| 5 | istpsi.j | . . . 4 ⊢ (TopOpen‘𝐾) = 𝐽 | |
| 6 | 5 | eqcomi 2746 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) |
| 7 | 4, 6 | istps2 22941 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
| 8 | 1, 2, 7 | mpbir2an 711 | 1 ⊢ 𝐾 ∈ TopSp |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ‘cfv 6561 Basecbs 17247 TopOpenctopn 17466 Topctop 22899 TopSpctps 22938 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-top 22900 df-topon 22917 df-topsp 22939 |
| This theorem is referenced by: indistps2 23019 |
| Copyright terms: Public domain | W3C validator |