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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 23033 | . . . 4 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})) |
| 3 | 2 | fveq1d 6908 | . . 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 | eqid 2737 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) = (𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥}) | |
| 6 | sseq1 4009 | . . . . 5 ⊢ (𝑦 = 𝑆 → (𝑦 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑥)) | |
| 7 | 6 | rabbidv 3444 | . . . 4 ⊢ (𝑦 = 𝑆 → {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
| 8 | 7 | inteqd 4951 | . . 3 ⊢ (𝑦 = 𝑆 → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥} = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
| 9 | 1 | topopn 22912 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
| 10 | elpw2g 5333 | . . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
| 11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
| 12 | 11 | biimpar 477 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋) |
| 13 | 1 | topcld 23043 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
| 14 | sseq2 4010 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋)) | |
| 15 | 14 | rspcev 3622 | . . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥) |
| 16 | 13, 15 | sylan 580 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥) |
| 17 | intexrab 5347 | . . . 4 ⊢ (∃𝑥 ∈ (Clsd‘𝐽)𝑆 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) | |
| 18 | 16, 17 | sylib 218 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} ∈ V) |
| 19 | 5, 8, 12, 18 | fvmptd3 7039 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑦 ⊆ 𝑥})‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
| 20 | 4, 19 | eqtrd 2777 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 ↦ cmpt 5225 ‘cfv 6561 Topctop 22899 Clsdccld 23024 clsccl 23026 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-cld 23027 df-cls 23029 |
| This theorem is referenced by: cldcls 23050 clscld 23055 clsf 23056 clsval2 23058 clsss 23062 sscls 23064 |
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