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Mirrors > Home > MPE Home > Th. List > perfcls | Structured version Visualization version Unicode 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 |
Ref | Expression |
---|---|
perfcls | ↾t Perf ↾t Perf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpcls.1 | . . . . 5 | |
2 | 1 | lpcls 21168 | . . . 4 |
3 | 2 | sseq2d 3633 | . . 3 |
4 | t1top 21134 | . . . . . 6 | |
5 | 1 | clslp 20952 | . . . . . 6 |
6 | 4, 5 | sylan 488 | . . . . 5 |
7 | 6 | sseq1d 3632 | . . . 4 |
8 | ssequn1 3783 | . . . . 5 | |
9 | ssun2 3777 | . . . . . 6 | |
10 | eqss 3618 | . . . . . 6 | |
11 | 9, 10 | mpbiran2 954 | . . . . 5 |
12 | 8, 11 | bitri 264 | . . . 4 |
13 | 7, 12 | syl6bbr 278 | . . 3 |
14 | 3, 13 | bitr2d 269 | . 2 |
15 | eqid 2622 | . . . 4 ↾t ↾t | |
16 | 1, 15 | restperf 20988 | . . 3 ↾t Perf |
17 | 4, 16 | sylan 488 | . 2 ↾t Perf |
18 | 1 | clsss3 20863 | . . . 4 |
19 | eqid 2622 | . . . . 5 ↾t ↾t | |
20 | 1, 19 | restperf 20988 | . . . 4 ↾t Perf |
21 | 18, 20 | syldan 487 | . . 3 ↾t Perf |
22 | 4, 21 | sylan 488 | . 2 ↾t Perf |
23 | 14, 17, 22 | 3bitr4d 300 | 1 ↾t Perf ↾t Perf |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cun 3572 wss 3574 cuni 4436 cfv 5888 (class class class)co 6650 ↾t crest 16081 ctop 20698 ccl 20822 clp 20938 Perfcperf 20939 ct1 21111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-t1 21118 |
This theorem is referenced by: (None) |
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