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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 21219 | . . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
4 | resttopon 21463 | . . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) | |
5 | 3, 4 | sylanb 573 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌)) |
6 | 1, 5 | syl5eqel 2864 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
7 | topontop 21215 | . . . 4 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top) |
9 | eqid 2772 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
10 | 9 | isperf 21453 | . . . 4 ⊢ (𝐾 ∈ Perf ↔ (𝐾 ∈ Top ∧ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
11 | 10 | baib 528 | . . 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 4074 | . . 3 ⊢ (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌) | |
14 | ssid 3875 | . . . . . 6 ⊢ 𝑌 ⊆ 𝑌 | |
15 | 2, 1 | restlp 21485 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋 ∧ 𝑌 ⊆ 𝑌) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
16 | 14, 15 | mp3an3 1429 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = (((limPt‘𝐽)‘𝑌) ∩ 𝑌)) |
17 | toponuni 21216 | . . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝐾) | |
18 | 6, 17 | syl 17 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → 𝑌 = ∪ 𝐾) |
19 | 18 | fveq2d 6497 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((limPt‘𝐾)‘𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
20 | 16, 19 | eqtr3d 2810 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (((limPt‘𝐽)‘𝑌) ∩ 𝑌) = ((limPt‘𝐾)‘∪ 𝐾)) |
21 | 20, 18 | eqeq12d 2787 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → ((((limPt‘𝐽)‘𝑌) ∩ 𝑌) = 𝑌 ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
22 | 13, 21 | syl5bb 275 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝑌 ⊆ ((limPt‘𝐽)‘𝑌) ↔ ((limPt‘𝐾)‘∪ 𝐾) = ∪ 𝐾)) |
23 | 12, 22 | bitr4d 274 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ 𝑋) → (𝐾 ∈ Perf ↔ 𝑌 ⊆ ((limPt‘𝐽)‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∩ cin 3824 ⊆ wss 3825 ∪ cuni 4706 ‘cfv 6182 (class class class)co 6970 ↾t crest 16540 Topctop 21195 TopOnctopon 21212 limPtclp 21436 Perfcperf 21437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-oadd 7901 df-er 8081 df-en 8299 df-fin 8302 df-fi 8662 df-rest 16542 df-topgen 16563 df-top 21196 df-topon 21213 df-bases 21248 df-cld 21321 df-cls 21323 df-lp 21438 df-perf 21439 |
This theorem is referenced by: perfcls 21667 reperflem 23119 perfdvf 24194 |
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