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Mirrors > Home > MPE Home > Th. List > restperf | Structured version Visualization version GIF version |
Description: Perfection of a subspace. Note that the term "perfect set" is reserved for closed sets which are perfect in the subspace topology. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
restcls.1 | ⊢ 𝑋 = ∪ 𝐽 |
restcls.2 | ⊢ 𝐾 = (𝐽 ↾t 𝑌) |
Ref | Expression |
---|---|
restperf | ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restcls.2 | . . . . 5 ⊢ 𝐾 = (𝐽 ↾t 𝑌) | |
2 | restcls.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | toptopon 21519 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
4 | resttopon 21763 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) | |
5 | 3, 4 | sylanb 583 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) |
6 | 1, 5 | eqeltrid 2917 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
7 | topontop 21515 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top) |
9 | eqid 2821 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
10 | 9 | isperf 21753 | . . . 4 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
11 | 10 | baib 538 | . . 3 ⊢ (𝐾 ∈ Top → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
12 | 8, 11 | syl 17 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
13 | sseqin2 4191 | . . 3 ⊢ (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌) | |
14 | ssid 3988 | . . . . . 6 ⊢ 𝑌 ⊆ 𝑌 | |
15 | 2, 1 | restlp 21785 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
16 | 14, 15 | mp3an3 1446 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
17 | toponuni 21516 | . . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
18 | 6, 17 | syl 17 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ 𝐾) |
19 | 18 | fveq2d 6668 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
20 | 16, 19 | eqtr3d 2858 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
21 | 20, 18 | eqeq12d 2837 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
22 | 13, 21 | syl5bb 285 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
23 | 12, 22 | bitr4d 284 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4831 ‘cfv 6349 (class class class)co 7150 ↾t crest 16688 Topctop 21495 TopOnctopon 21512 limPtclp 21736 Perfcperf 21737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-rest 16690 df-topgen 16711 df-top 21496 df-topon 21513 df-bases 21548 df-cld 21621 df-cls 21623 df-lp 21738 df-perf 21739 |
This theorem is referenced by: perfcls 21967 reperflem 23420 perfdvf 24495 |
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