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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 2724 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 2 | cldss 22855 | . . 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 4014 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
9 | eqid 2724 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | restuni 22988 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
11 | 5, 8, 10 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
12 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
13 | 12 | unieqd 4912 | . . 3 ⊢ (𝜑 → ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)) |
14 | 11, 13 | eqtr4d 2767 | . 2 ⊢ (𝜑 → 𝑌 = ∪ 𝐾) |
15 | 4, 14 | sseqtrrd 4015 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 ∪ cuni 4899 ‘cfv 6533 (class class class)co 7401 ↾t crest 17365 Topctop 22717 Clsdccld 22842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-en 8936 df-fin 8939 df-fi 9402 df-rest 17367 df-topgen 17388 df-top 22718 df-topon 22735 df-bases 22771 df-cld 22845 |
This theorem is referenced by: restcls2 47734 restclssep 47736 iscnrm3llem1 47770 iscnrm3llem2 47771 |
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