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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 4896 | . 2 ⊢ 𝑆 ⊆ ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} | |
2 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval 21647 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥}) |
4 | 1, 3 | sseqtrrid 4022 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ⊆ wss 3938 ∪ cuni 4840 ∩ cint 4878 ‘cfv 6357 Topctop 21503 Clsdccld 21626 clsccl 21628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-top 21504 df-cld 21629 df-cls 21631 |
This theorem is referenced by: iscld4 21675 elcls 21683 ntrcls0 21686 clslp 21758 restcls 21791 cncls2i 21880 nrmsep 21967 lpcls 21974 regsep2 21986 hauscmplem 22016 hauscmp 22017 clsconn 22040 conncompcld 22044 hausllycmp 22104 txcls 22214 ptclsg 22225 regr1lem 22349 kqreglem1 22351 kqreglem2 22352 kqnrmlem1 22353 kqnrmlem2 22354 fclscmpi 22639 flfcntr 22653 cnextfres 22679 clssubg 22719 tsmsid 22750 cnllycmp 23562 clsocv 23855 relcmpcmet 23923 bcthlem2 23930 bcthlem4 23932 limcnlp 24478 opnbnd 33675 opnregcld 33680 cldregopn 33681 heibor1lem 35089 heiborlem8 35098 |
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