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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 20548 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
4 | resttopon 20775 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) | |
5 | 3, 4 | sylanb 488 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) |
6 | 1, 5 | syl5eqel 2692 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
7 | topontop 20541 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top) |
9 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
10 | 9 | isperf 20765 | . . . 4 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
11 | 10 | baib 942 | . . 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 3779 | . . 3 ⊢ (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌) | |
14 | ssid 3587 | . . . . . 6 ⊢ 𝑌 ⊆ 𝑌 | |
15 | 2, 1 | restlp 20797 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
16 | 14, 15 | mp3an3 1405 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
17 | toponuni 20542 | . . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
18 | 6, 17 | syl 17 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ 𝐾) |
19 | 18 | fveq2d 6107 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
20 | 16, 19 | eqtr3d 2646 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
21 | 20, 18 | eqeq12d 2625 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
22 | 13, 21 | syl5bb 271 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
23 | 12, 22 | bitr4d 270 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 (class class class)co 6549 ↾t crest 15904 Topctop 20517 TopOnctopon 20518 limPtclp 20748 Perfcperf 20749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-cls 20635 df-lp 20750 df-perf 20751 |
This theorem is referenced by: perfcls 20979 reperflem 22429 perfdvf 23473 |
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