![]() |
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 22543 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽)) |
3 | 1 | cldss 22525 | . 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 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3948 ∪ cuni 4908 ‘cfv 6541 Topctop 22387 Clsdccld 22512 clsccl 22514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-top 22388 df-cld 22515 df-cls 22517 |
This theorem is referenced by: clsidm 22563 elcls2 22570 clsndisj 22571 ntrcls0 22572 neindisj 22613 lpval 22635 lpss 22638 clslp 22644 cnclsi 22768 cncls 22770 isnrm2 22854 lpcls 22860 perfcls 22861 regsep2 22872 clsconn 22926 conncompcld 22930 2ndcsep 22955 1stcelcls 22957 hausllycmp 22990 txcls 23100 ptclsg 23111 imasncls 23188 kqnrmlem1 23239 reghmph 23289 nrmhmph 23290 flimclslem 23480 flimsncls 23482 hauspwpwf1 23483 fclsopn 23510 fclscmpi 23525 cnextfun 23560 clssubg 23605 clsnsg 23606 snclseqg 23612 utop3cls 23748 qdensere 24278 clsocv 24759 relcmpcmet 24827 cncmet 24831 kur14lem3 34188 topbnd 35198 clsun 35202 opnregcld 35204 cldregopn 35205 heibor1lem 36666 qndenserrn 45002 iscnrm3rlem2 47528 |
Copyright terms: Public domain | W3C validator |