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Mirrors > Home > MPE Home > Th. List > clscld | Structured version Visualization version GIF version |
Description: The closure of a subset of a topology's underlying set is closed. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
clscld | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | clsval 22540 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
3 | 1 | topcld 22538 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
4 | 3 | anim1i 615 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋)) |
5 | sseq2 4008 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋)) | |
6 | 5 | elrab 3683 | . . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋)) |
7 | 4, 6 | sylibr 233 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
8 | 7 | ne0d 4335 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅) |
9 | ssrab2 4077 | . . 3 ⊢ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽) | |
10 | intcld 22543 | . . 3 ⊢ (({𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ⊆ (Clsd‘𝐽)) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽)) | |
11 | 8, 9, 10 | sylancl 586 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ (Clsd‘𝐽)) |
12 | 2, 11 | eqeltrd 2833 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 ∩ cint 4950 ‘cfv 6543 Topctop 22394 Clsdccld 22519 clsccl 22521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-top 22395 df-cld 22522 df-cls 22524 |
This theorem is referenced by: clsf 22551 clsss3 22562 iscld3 22567 clsidm 22570 restcls 22684 cncls2i 22773 nrmsep 22860 lpcls 22867 regsep2 22879 hauscmplem 22909 hausllycmp 22997 txcls 23107 ptclsg 23118 regr1lem 23242 kqreglem1 23244 kqreglem2 23245 kqnrmlem1 23246 kqnrmlem2 23247 fclscmpi 23532 tgptsmscld 23654 cnllycmp 24471 clsocv 24766 cmpcmet 24835 cncmet 24838 limcnlp 25394 clsun 35208 cldregopn 35211 heibor1lem 36672 iscnrm3rlem2 47564 iscnrm3rlem5 47567 |
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