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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 2730 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 2 | cldss 22753 | . . 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 4021 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
9 | eqid 2730 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | restuni 22886 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
11 | 5, 8, 10 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
12 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
13 | 12 | unieqd 4921 | . . 3 ⊢ (𝜑 → ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)) |
14 | 11, 13 | eqtr4d 2773 | . 2 ⊢ (𝜑 → 𝑌 = ∪ 𝐾) |
15 | 4, 14 | sseqtrrd 4022 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2104 ⊆ wss 3947 ∪ cuni 4907 ‘cfv 6542 (class class class)co 7411 ↾t crest 17370 Topctop 22615 Clsdccld 22740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17372 df-topgen 17393 df-top 22616 df-topon 22633 df-bases 22669 df-cld 22743 |
This theorem is referenced by: restcls2 47633 restclssep 47635 iscnrm3llem1 47669 iscnrm3llem2 47670 |
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