Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4897 | . 2 ⊢ 𝑆 ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} | |
2 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 22188 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
4 | 1, 3 | sseqtrrid 3974 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 ⊆ wss 3887 ∪ cuni 4839 ∩ cint 4879 ‘cfv 6433 Topctop 22042 Clsdccld 22167 clsccl 22169 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-top 22043 df-cld 22170 df-cls 22172 |
This theorem is referenced by: iscld4 22216 elcls 22224 ntrcls0 22227 clslp 22299 restcls 22332 cncls2i 22421 nrmsep 22508 lpcls 22515 regsep2 22527 hauscmplem 22557 hauscmp 22558 clsconn 22581 conncompcld 22585 hausllycmp 22645 txcls 22755 ptclsg 22766 regr1lem 22890 kqreglem1 22892 kqreglem2 22893 kqnrmlem1 22894 kqnrmlem2 22895 fclscmpi 23180 flfcntr 23194 cnextfres 23220 clssubg 23260 tsmsid 23291 cnllycmp 24119 clsocv 24414 relcmpcmet 24482 bcthlem2 24489 bcthlem4 24491 limcnlp 25042 opnbnd 34514 opnregcld 34519 cldregopn 34520 heibor1lem 35967 heiborlem8 35976 sepdisj 46218 iscnrm3rlem4 46237 |
Copyright terms: Public domain | W3C validator |