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Mirrors > Home > MPE Home > Th. List > clsval | Structured version Visualization version GIF version |
Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
clsval | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | clsfval 20877 | . . . 4 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})) |
3 | 2 | fveq1d 6231 | . . 3 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆)) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆)) |
5 | 1 | topopn 20759 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
6 | elpw2g 4857 | . . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
8 | 7 | biimpar 501 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋) |
9 | 1 | topcld 20887 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
10 | sseq2 3660 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋)) | |
11 | 10 | rspcev 3340 | . . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥) |
12 | 9, 11 | sylan 487 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥) |
13 | intexrab 4853 | . . . 4 ⊢ (∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) | |
14 | 12, 13 | sylib 208 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) |
15 | sseq1 3659 | . . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑥)) | |
16 | 15 | rabbidv 3220 | . . . . 5 ⊢ (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
17 | 16 | inteqd 4512 | . . . 4 ⊢ (𝑦 = 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
18 | eqid 2651 | . . . 4 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) | |
19 | 17, 18 | fvmptg 6319 | . . 3 ⊢ ((𝑆 ∈ 𝒫 𝑋 ∧ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
20 | 8, 14, 19 | syl2anc 694 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
21 | 4, 20 | eqtrd 2685 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∃wrex 2942 {crab 2945 Vcvv 3231 ⊆ wss 3607 𝒫 cpw 4191 ∪ cuni 4468 ∩ cint 4507 ↦ cmpt 4762 ‘cfv 5926 Topctop 20746 Clsdccld 20868 clsccl 20870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-top 20747 df-cld 20871 df-cls 20873 |
This theorem is referenced by: cldcls 20894 clscld 20899 clsf 20900 clsval2 20902 clsss 20906 sscls 20908 |
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