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| Mirrors > Home > MPE Home > Th. List > Mathboxes > restcls2lem | Structured version Visualization version GIF version | ||
| Description: A closed set in a subspace topology is a subset of the subspace. (Contributed by Zhi Wang, 2-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| restcls2.1 | ⊢ (𝜑 → 𝐽 ∈ Top) | 
| restcls2.2 | ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | 
| restcls2.3 | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | 
| restcls2.4 | ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | 
| restcls2.5 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | 
| Ref | Expression | 
|---|---|
| restcls2lem | ⊢ (𝜑 → 𝑆 ⊆ 𝑌) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | restcls2.5 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | |
| 2 | eqid 2736 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
| 3 | 2 | cldss 23038 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐾) → 𝑆 ⊆ ∪ 𝐾) | 
| 4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐾) | 
| 5 | restcls2.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | |
| 6 | restcls2.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
| 7 | restcls2.2 | . . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | |
| 8 | 6, 7 | sseqtrd 4019 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) | 
| 9 | eqid 2736 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 10 | 9 | restuni 23171 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) | 
| 11 | 5, 8, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 ↾t 𝑌)) | 
| 12 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
| 13 | 12 | unieqd 4919 | . . 3 ⊢ (𝜑 → ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)) | 
| 14 | 11, 13 | eqtr4d 2779 | . 2 ⊢ (𝜑 → 𝑌 = ∪ 𝐾) | 
| 15 | 4, 14 | sseqtrrd 4020 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ∪ cuni 4906 ‘cfv 6560 (class class class)co 7432 ↾t crest 17466 Topctop 22900 Clsdccld 23025 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-en 8987 df-fin 8990 df-fi 9452 df-rest 17468 df-topgen 17489 df-top 22901 df-topon 22918 df-bases 22954 df-cld 23028 | 
| This theorem is referenced by: restcls2 48818 restclssep 48820 iscnrm3llem1 48853 iscnrm3llem2 48854 | 
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