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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 21636 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐽 ∈ Top) |
3 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | cldss 21639 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋) |
5 | 4 | adantr 483 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐶 ⊆ 𝑋) |
6 | simpr 487 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ⊆ 𝐶) | |
7 | 3 | clsss 21664 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
8 | 2, 5, 6, 7 | syl3anc 1367 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
9 | cldcls 21652 | . . 3 ⊢ (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶) | |
10 | 9 | adantr 483 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶) |
11 | 8, 10 | sseqtrd 4009 | 1 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 ‘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 ax-un 7463 |
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: elcls 21683 restcls 21791 cncls2i 21880 isnrm3 21969 lpcls 21974 isreg2 21987 dnsconst 21988 hauscmplem 22016 txcls 22214 ptclsg 22225 kqreglem1 22351 kqreglem2 22352 kqnrmlem1 22353 kqnrmlem2 22354 blcls 23118 clsocv 23855 resscdrg 23963 cldregopn 33681 |
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