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 4430 | . 2 ⊢ 𝑆 ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} | |
2 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 20651 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
4 | 1, 3 | syl5sseqr 3617 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 ∪ cuni 4372 ∩ cint 4410 ‘cfv 5804 Topctop 20517 Clsdccld 20630 clsccl 20632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-top 20521 df-cld 20633 df-cls 20635 |
This theorem is referenced by: iscld4 20679 elcls 20687 ntrcls0 20690 clslp 20762 restcls 20795 cncls2i 20884 nrmsep 20971 lpcls 20978 regsep2 20990 hauscmplem 21019 hauscmp 21020 clscon 21043 concompcld 21047 hausllycmp 21107 txcls 21217 ptclsg 21228 regr1lem 21352 kqreglem1 21354 kqreglem2 21355 kqnrmlem1 21356 kqnrmlem2 21357 fclscmpi 21643 flfcntr 21657 cnextfres 21683 clssubg 21722 tsmsid 21753 cnllycmp 22563 clsocv 22857 relcmpcmet 22923 bcthlem2 22930 bcthlem4 22932 limcnlp 23448 opnbnd 31490 opnregcld 31495 cldregopn 31496 heibor1lem 32778 heiborlem8 32787 |
Copyright terms: Public domain | W3C validator |