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Mirrors > Home > MPE Home > Th. List > clsss2 | Structured version Visualization version GIF version |
Description: If a subset is included in a closed set, so is the subset's closure. (Contributed by NM, 22-Feb-2007.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
clsss2 | ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldrcl 21159 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | 1 | adantr 473 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐽 ∈ Top) |
3 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | cldss 21162 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋) |
5 | 4 | adantr 473 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐶 ⊆ 𝑋) |
6 | simpr 478 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ⊆ 𝐶) | |
7 | 3 | clsss 21187 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
8 | 2, 5, 6, 7 | syl3anc 1491 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
9 | cldcls 21175 | . . 3 ⊢ (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶) | |
10 | 9 | adantr 473 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶) |
11 | 8, 10 | sseqtrd 3837 | 1 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 ∪ cuni 4628 ‘cfv 6101 Topctop 21026 Clsdccld 21149 clsccl 21151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-top 21027 df-cld 21152 df-cls 21154 |
This theorem is referenced by: elcls 21206 restcls 21314 cncls2i 21403 isnrm3 21492 lpcls 21497 isreg2 21510 dnsconst 21511 hauscmplem 21538 txcls 21736 ptclsg 21747 kqreglem1 21873 kqreglem2 21874 kqnrmlem1 21875 kqnrmlem2 21876 blcls 22639 clsocv 23376 resscdrg 23484 cldregopn 32838 |
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