|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > clsss3 | Structured version Visualization version GIF version | ||
| Description: The closure of a subset of a topological space is included in the space. (Contributed by NM, 26-Feb-2007.) | 
| Ref | Expression | 
|---|---|
| clscld.1 | ⊢ 𝑋 = ∪ 𝐽 | 
| Ref | Expression | 
|---|---|
| clsss3 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | clscld 23056 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) | 
| 3 | 1 | cldss 23038 | . 2 ⊢ (((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | 
| 4 | 2, 3 | syl 17 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 ‘cfv 6560 Topctop 22900 Clsdccld 23025 clsccl 23027 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-top 22901 df-cld 23028 df-cls 23030 | 
| This theorem is referenced by: clsidm 23076 elcls2 23083 clsndisj 23084 ntrcls0 23085 neindisj 23126 lpval 23148 lpss 23151 clslp 23157 cnclsi 23281 cncls 23283 isnrm2 23367 lpcls 23373 perfcls 23374 regsep2 23385 clsconn 23439 conncompcld 23443 2ndcsep 23468 1stcelcls 23470 hausllycmp 23503 txcls 23613 ptclsg 23624 imasncls 23701 kqnrmlem1 23752 reghmph 23802 nrmhmph 23803 flimclslem 23993 flimsncls 23995 hauspwpwf1 23996 fclsopn 24023 fclscmpi 24038 cnextfun 24073 clssubg 24118 clsnsg 24119 snclseqg 24125 utop3cls 24261 qdensere 24791 clsocv 25285 relcmpcmet 25353 cncmet 25357 kur14lem3 35214 topbnd 36326 clsun 36330 opnregcld 36332 cldregopn 36333 heibor1lem 37817 qndenserrn 46319 iscnrm3rlem2 48845 | 
| Copyright terms: Public domain | W3C validator |