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Mirrors > Home > MPE Home > Th. List > Mathboxes > clduni | Structured version Visualization version GIF version |
Description: The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.) |
Ref | Expression |
---|---|
clduni | ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 22922 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | 1 | biimpi 216 | . 2 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
3 | cldmreon 23100 | . 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) | |
4 | mreuni 17635 | . 2 ⊢ ((Clsd‘𝐽) ∈ (Moore‘∪ 𝐽) → ∪ (Clsd‘𝐽) = ∪ 𝐽) | |
5 | 2, 3, 4 | 3syl 18 | 1 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ∪ cuni 4915 ‘cfv 6559 Moorecmre 17617 Topctop 22897 TopOnctopon 22914 Clsdccld 23022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7748 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6511 df-fun 6561 df-fn 6562 df-fv 6567 df-mre 17621 df-top 22898 df-topon 22915 df-cld 23025 |
This theorem is referenced by: clddisj 48628 |
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