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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 23050 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐽 ∈ Top) |
3 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | cldss 23053 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋) |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐶 ⊆ 𝑋) |
6 | simpr 484 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ⊆ 𝐶) | |
7 | 3 | clsss 23078 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
8 | 2, 5, 6, 7 | syl3anc 1370 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
9 | cldcls 23066 | . . 3 ⊢ (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶) | |
10 | 9 | adantr 480 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶) |
11 | 8, 10 | sseqtrd 4036 | 1 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 ‘cfv 6563 Topctop 22915 Clsdccld 23040 clsccl 23042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-top 22916 df-cld 23043 df-cls 23045 |
This theorem is referenced by: elcls 23097 restcls 23205 cncls2i 23294 isnrm3 23383 lpcls 23388 isreg2 23401 dnsconst 23402 hauscmplem 23430 txcls 23628 ptclsg 23639 kqreglem1 23765 kqreglem2 23766 kqnrmlem1 23767 kqnrmlem2 23768 blcls 24535 clsocv 25298 resscdrg 25406 cldregopn 36314 seposep 48722 |
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