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Mirrors > Home > MPE Home > Th. List > lpss3 | Structured version Visualization version GIF version |
Description: Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
lpss3 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝐽 ∈ Top) | |
2 | simp2 1136 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑆 ⊆ 𝑋) | |
3 | 2 | ssdifssd 4156 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
4 | simp3 1137 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑆) | |
5 | 4 | ssdifd 4154 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
6 | lpfval.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | clsss 23077 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑥}) ⊆ 𝑋 ∧ (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) |
8 | 1, 3, 5, 7 | syl3anc 1370 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) ⊆ ((cls‘𝐽)‘(𝑆 ∖ {𝑥}))) |
9 | 8 | sseld 3993 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})) → 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
10 | 4, 2 | sstrd 4005 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ 𝑋) |
11 | 6 | islp 23163 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑇 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})))) |
12 | 1, 10, 11 | syl2anc 584 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑇 ∖ {𝑥})))) |
13 | 6 | islp 23163 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
14 | 1, 2, 13 | syl2anc 584 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑥 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑥})))) |
15 | 9, 12, 14 | 3imtr4d 294 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → (𝑥 ∈ ((limPt‘𝐽)‘𝑇) → 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
16 | 15 | ssrdv 4000 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑇 ⊆ 𝑆) → ((limPt‘𝐽)‘𝑇) ⊆ ((limPt‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 ∪ cuni 4911 ‘cfv 6562 Topctop 22914 clsccl 23041 limPtclp 23157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-top 22915 df-cld 23042 df-cls 23044 df-lp 23159 |
This theorem is referenced by: perfdvf 25952 pibt2 37399 lpss2 37740 fourierdlem113 46174 |
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