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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 | β’ π = βͺ π½ |
restcls.2 | β’ πΎ = (π½ βΎt π) |
Ref | Expression |
---|---|
restperf | β’ ((π½ β Top β§ π β π) β (πΎ β Perf β π β ((limPtβπ½)βπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restcls.2 | . . . . 5 β’ πΎ = (π½ βΎt π) | |
2 | restcls.1 | . . . . . . 7 β’ π = βͺ π½ | |
3 | 2 | toptopon 22774 | . . . . . 6 β’ (π½ β Top β π½ β (TopOnβπ)) |
4 | resttopon 23020 | . . . . . 6 β’ ((π½ β (TopOnβπ) β§ π β π) β (π½ βΎt π) β (TopOnβπ)) | |
5 | 3, 4 | sylanb 580 | . . . . 5 β’ ((π½ β Top β§ π β π) β (π½ βΎt π) β (TopOnβπ)) |
6 | 1, 5 | eqeltrid 2831 | . . . 4 β’ ((π½ β Top β§ π β π) β πΎ β (TopOnβπ)) |
7 | topontop 22770 | . . . 4 β’ (πΎ β (TopOnβπ) β πΎ β Top) | |
8 | 6, 7 | syl 17 | . . 3 β’ ((π½ β Top β§ π β π) β πΎ β Top) |
9 | eqid 2726 | . . . . 5 β’ βͺ πΎ = βͺ πΎ | |
10 | 9 | isperf 23010 | . . . 4 β’ (πΎ β Perf β (πΎ β Top β§ ((limPtβπΎ)ββͺ πΎ) = βͺ πΎ)) |
11 | 10 | baib 535 | . . 3 β’ (πΎ β Top β (πΎ β Perf β ((limPtβπΎ)ββͺ πΎ) = βͺ πΎ)) |
12 | 8, 11 | syl 17 | . 2 β’ ((π½ β Top β§ π β π) β (πΎ β Perf β ((limPtβπΎ)ββͺ πΎ) = βͺ πΎ)) |
13 | sseqin2 4210 | . . 3 β’ (π β ((limPtβπ½)βπ) β (((limPtβπ½)βπ) β© π) = π) | |
14 | ssid 3999 | . . . . . 6 β’ π β π | |
15 | 2, 1 | restlp 23042 | . . . . . 6 β’ ((π½ β Top β§ π β π β§ π β π) β ((limPtβπΎ)βπ) = (((limPtβπ½)βπ) β© π)) |
16 | 14, 15 | mp3an3 1446 | . . . . 5 β’ ((π½ β Top β§ π β π) β ((limPtβπΎ)βπ) = (((limPtβπ½)βπ) β© π)) |
17 | toponuni 22771 | . . . . . . 7 β’ (πΎ β (TopOnβπ) β π = βͺ πΎ) | |
18 | 6, 17 | syl 17 | . . . . . 6 β’ ((π½ β Top β§ π β π) β π = βͺ πΎ) |
19 | 18 | fveq2d 6889 | . . . . 5 β’ ((π½ β Top β§ π β π) β ((limPtβπΎ)βπ) = ((limPtβπΎ)ββͺ πΎ)) |
20 | 16, 19 | eqtr3d 2768 | . . . 4 β’ ((π½ β Top β§ π β π) β (((limPtβπ½)βπ) β© π) = ((limPtβπΎ)ββͺ πΎ)) |
21 | 20, 18 | eqeq12d 2742 | . . 3 β’ ((π½ β Top β§ π β π) β ((((limPtβπ½)βπ) β© π) = π β ((limPtβπΎ)ββͺ πΎ) = βͺ πΎ)) |
22 | 13, 21 | bitrid 283 | . 2 β’ ((π½ β Top β§ π β π) β (π β ((limPtβπ½)βπ) β ((limPtβπΎ)ββͺ πΎ) = βͺ πΎ)) |
23 | 12, 22 | bitr4d 282 | 1 β’ ((π½ β Top β§ π β π) β (πΎ β Perf β π β ((limPtβπ½)βπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 β© cin 3942 β wss 3943 βͺ cuni 4902 βcfv 6537 (class class class)co 7405 βΎt crest 17375 Topctop 22750 TopOnctopon 22767 limPtclp 22993 Perfcperf 22994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17377 df-topgen 17398 df-top 22751 df-topon 22768 df-bases 22804 df-cld 22878 df-cls 22880 df-lp 22995 df-perf 22996 |
This theorem is referenced by: perfcls 23224 reperflem 24689 perfdvf 25787 |
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