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| Mirrors > Home > MPE Home > Th. List > lpval | Structured version Visualization version GIF version | ||
| Description: The set of limit points of a subset of the base set of a topology. Alternate definition of limit point in [Munkres] p. 97. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| lpval | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lpfval.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | lpfval 23146 | . . . 4 ⊢ (𝐽 ∈ Top → (limPt‘𝐽) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})) |
| 3 | 2 | fveq1d 6908 | . . 3 ⊢ (𝐽 ∈ Top → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆)) |
| 4 | 3 | adantr 480 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆)) |
| 5 | eqid 2737 | . . 3 ⊢ (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) = (𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))}) | |
| 6 | difeq1 4119 | . . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ∖ {𝑥}) = (𝑆 ∖ {𝑥})) | |
| 7 | 6 | fveq2d 6910 | . . . . 5 ⊢ (𝑦 = 𝑆 → ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) = ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) |
| 8 | 7 | eleq2d 2827 | . . . 4 ⊢ (𝑦 = 𝑆 → (𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥})) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
| 9 | 8 | abbidv 2808 | . . 3 ⊢ (𝑦 = 𝑆 → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))} = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}) |
| 10 | 1 | topopn 22912 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
| 11 | elpw2g 5333 | . . . . 5 ⊢ (𝑋 ∈ 𝐽 → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
| 12 | 10, 11 | syl 17 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
| 13 | 12 | biimpar 477 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋) |
| 14 | 10 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ 𝐽) |
| 15 | ssdifss 4140 | . . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) | |
| 16 | 1 | clsss3 23067 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) ⊆ 𝑋) |
| 17 | 16 | sseld 3982 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ 𝑋)) |
| 18 | 15, 17 | sylan2 593 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})) → 𝑥 ∈ 𝑋)) |
| 19 | 18 | abssdv 4068 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ⊆ 𝑋) |
| 20 | 14, 19 | ssexd 5324 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))} ∈ V) |
| 21 | 5, 9, 13, 20 | fvmptd3 7039 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑦 ∈ 𝒫 𝑋 ↦ {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑦 ∖ {𝑥}))})‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}) |
| 22 | 4, 21 | eqtrd 2777 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) = {𝑥 ∣ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ‘cfv 6561 Topctop 22899 clsccl 23026 limPtclp 23142 |
| 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 ax-un 7755 |
| 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-iin 4994 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 df-lp 23144 |
| This theorem is referenced by: islp 23148 lpsscls 23149 |
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