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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 21562 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐽 ∈ Top) |
3 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | cldss 21565 | . . . 4 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐶 ⊆ 𝑋) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝐶 ⊆ 𝑋) |
6 | simpr 485 | . . 3 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → 𝑆 ⊆ 𝐶) | |
7 | 3 | clsss 21590 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐶 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
8 | 2, 5, 6, 7 | syl3anc 1363 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝐶)) |
9 | cldcls 21578 | . . 3 ⊢ (𝐶 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝐶) = 𝐶) | |
10 | 9 | adantr 481 | . 2 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝐶) = 𝐶) |
11 | 8, 10 | sseqtrd 4004 | 1 ⊢ ((𝐶 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝐶) → ((cls‘𝐽)‘𝑆) ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ∪ cuni 4830 ‘cfv 6348 Topctop 21429 Clsdccld 21552 clsccl 21554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-top 21430 df-cld 21555 df-cls 21557 |
This theorem is referenced by: elcls 21609 restcls 21717 cncls2i 21806 isnrm3 21895 lpcls 21900 isreg2 21913 dnsconst 21914 hauscmplem 21942 txcls 22140 ptclsg 22151 kqreglem1 22277 kqreglem2 22278 kqnrmlem1 22279 kqnrmlem2 22280 blcls 23043 clsocv 23780 resscdrg 23888 cldregopn 33576 |
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