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Mirrors > Home > MPE Home > Th. List > cldmreon | Structured version Visualization version GIF version |
Description: The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
cldmreon | ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 22934 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
2 | eqid 2734 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | cldmre 23101 | . . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽)) |
5 | toponuni 22935 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
6 | 5 | fveq2d 6910 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘∪ 𝐽)) |
7 | 4, 6 | eleqtrrd 2841 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cuni 4911 ‘cfv 6562 Moorecmre 17626 Topctop 22914 TopOnctopon 22931 Clsdccld 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 df-mre 17630 df-top 22915 df-topon 22932 df-cld 23042 |
This theorem is referenced by: iscldtop 23118 clduni 48696 |
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