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Mirrors > Home > MPE Home > Th. List > cldcls | Structured version Visualization version GIF version |
Description: A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.) |
Ref | Expression |
---|---|
cldcls | ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldrcl 21626 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | eqid 2819 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | 2 | cldss 21629 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
4 | 2 | clsval 21637 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
5 | 1, 3, 4 | syl2anc 586 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
6 | intmin 4887 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} = 𝑆) | |
7 | 5, 6 | eqtrd 2854 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 {crab 3140 ⊆ wss 3934 ∪ cuni 4830 ∩ cint 4867 ‘cfv 6348 Topctop 21493 Clsdccld 21616 clsccl 21618 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-top 21494 df-cld 21619 df-cls 21621 |
This theorem is referenced by: iscld3 21664 clstop 21669 clsss2 21672 cls0 21680 cncls2 21873 lmcld 21903 fclscmp 22630 metnrmlem1a 23458 lebnumlem1 23557 cmetss 23911 minveclem4 24027 hauseqcn 31131 |
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