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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 2631 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
5 | istpsi.j | . . . 4 ⊢ (TopOpen‘𝐾) = 𝐽 | |
6 | 5 | eqcomi 2631 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) |
7 | 4, 6 | istps2 20739 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
8 | 1, 2, 7 | mpbir2an 955 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 ∪ cuni 4436 ‘cfv 5888 Basecbs 15857 TopOpenctopn 16082 Topctop 20698 TopSpctps 20736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-topon 20716 df-topsp 20737 |
This theorem is referenced by: indistps2 20816 |
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