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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 21652 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
3 | 1 | cldss 21634 | . 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 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 Clsdccld 21621 clsccl 21623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 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 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-top 21499 df-cld 21624 df-cls 21626 |
This theorem is referenced by: clsidm 21672 elcls2 21679 clsndisj 21680 ntrcls0 21681 neindisj 21722 lpval 21744 lpss 21747 clslp 21753 cnclsi 21877 cncls 21879 isnrm2 21963 lpcls 21969 perfcls 21970 regsep2 21981 clsconn 22035 conncompcld 22039 2ndcsep 22064 1stcelcls 22066 hausllycmp 22099 txcls 22209 ptclsg 22220 imasncls 22297 kqnrmlem1 22348 reghmph 22398 nrmhmph 22399 flimclslem 22589 flimsncls 22591 hauspwpwf1 22592 fclsopn 22619 fclscmpi 22634 cnextfun 22669 clssubg 22714 clsnsg 22715 snclseqg 22721 utop3cls 22857 qdensere 23375 clsocv 23854 relcmpcmet 23922 cncmet 23926 kur14lem3 32568 topbnd 33785 clsun 33789 opnregcld 33791 cldregopn 33792 heibor1lem 35247 qndenserrn 42941 |
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