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Mirrors > Home > MPE Home > Th. List > perfcls | Structured version Visualization version GIF version |
Description: A subset of a perfect space is perfect iff its closure is perfect (and the closure is an actual perfect set, since it is both closed and perfect in the subspace topology). (Contributed by Mario Carneiro, 26-Dec-2016.) |
Ref | Expression |
---|---|
lpcls.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
perfcls | ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpcls.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | lpcls 23187 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆)) |
3 | 2 | sseq2d 4014 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆))) |
4 | t1top 23153 | . . . . . 6 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
5 | 1 | clslp 22971 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
6 | 4, 5 | sylan 579 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
7 | 6 | sseq1d 4013 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆))) |
8 | ssequn1 4180 | . . . . 5 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆)) | |
9 | ssun2 4173 | . . . . . 6 ⊢ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) | |
10 | eqss 3997 | . . . . . 6 ⊢ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆) ∧ ((limPt‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))) | |
11 | 9, 10 | mpbiran2 707 | . . . . 5 ⊢ ((𝑆 ∪ ((limPt‘𝐽)‘𝑆)) = ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)) |
12 | 8, 11 | bitri 275 | . . . 4 ⊢ (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((limPt‘𝐽)‘𝑆)) |
13 | 7, 12 | bitr4di 289 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆))) |
14 | 3, 13 | bitr2d 280 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → (𝑆 ⊆ ((limPt‘𝐽)‘𝑆) ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)))) |
15 | eqid 2731 | . . . 4 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t 𝑆) | |
16 | 1, 15 | restperf 23007 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆))) |
17 | 4, 16 | sylan 579 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ 𝑆 ⊆ ((limPt‘𝐽)‘𝑆))) |
18 | 1 | clsss3 22882 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) |
19 | eqid 2731 | . . . . 5 ⊢ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) = (𝐽 ↾t ((cls‘𝐽)‘𝑆)) | |
20 | 1, 19 | restperf 23007 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ((cls‘𝐽)‘𝑆) ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)))) |
21 | 18, 20 | syldan 590 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)))) |
22 | 4, 21 | sylan 579 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf ↔ ((cls‘𝐽)‘𝑆) ⊆ ((limPt‘𝐽)‘((cls‘𝐽)‘𝑆)))) |
23 | 14, 17, 22 | 3bitr4d 311 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝑆 ⊆ 𝑋) → ((𝐽 ↾t 𝑆) ∈ Perf ↔ (𝐽 ↾t ((cls‘𝐽)‘𝑆)) ∈ Perf)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∪ cun 3946 ⊆ wss 3948 ∪ cuni 4908 ‘cfv 6543 (class class class)co 7412 ↾t crest 17373 Topctop 22714 clsccl 22841 limPtclp 22957 Perfcperf 22958 Frect1 23130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 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-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 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-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-en 8946 df-fin 8949 df-fi 9412 df-rest 17375 df-topgen 17396 df-top 22715 df-topon 22732 df-bases 22768 df-cld 22842 df-ntr 22843 df-cls 22844 df-nei 22921 df-lp 22959 df-perf 22960 df-t1 23137 |
This theorem is referenced by: (None) |
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