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Mirrors > Home > MPE Home > Th. List > sscls | Structured version Visualization version GIF version |
Description: A subset of a topology's underlying set is included in its closure. (Contributed by NM, 22-Feb-2007.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
sscls | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssintub 4851 | . 2 ⊢ 𝑆 ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} | |
2 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 21781 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
4 | 1, 3 | sseqtrrid 3928 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {crab 3057 ⊆ wss 3841 ∪ cuni 4793 ∩ cint 4833 ‘cfv 6333 Topctop 21637 Clsdccld 21760 clsccl 21762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-top 21638 df-cld 21763 df-cls 21765 |
This theorem is referenced by: iscld4 21809 elcls 21817 ntrcls0 21820 clslp 21892 restcls 21925 cncls2i 22014 nrmsep 22101 lpcls 22108 regsep2 22120 hauscmplem 22150 hauscmp 22151 clsconn 22174 conncompcld 22178 hausllycmp 22238 txcls 22348 ptclsg 22359 regr1lem 22483 kqreglem1 22485 kqreglem2 22486 kqnrmlem1 22487 kqnrmlem2 22488 fclscmpi 22773 flfcntr 22787 cnextfres 22813 clssubg 22853 tsmsid 22884 cnllycmp 23701 clsocv 23995 relcmpcmet 24063 bcthlem2 24070 bcthlem4 24072 limcnlp 24622 opnbnd 34144 opnregcld 34149 cldregopn 34150 heibor1lem 35579 heiborlem8 35588 sepdisj 45724 iscnrm3rlem4 45743 |
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