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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 2758 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 2 | cldss 21742 | . . 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 3934 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
9 | eqid 2758 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | restuni 21875 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽) → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
11 | 5, 8, 10 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝑌 = ∪ (𝐽 ↾t 𝑌)) |
12 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
13 | 12 | unieqd 4815 | . . 3 ⊢ (𝜑 → ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)) |
14 | 11, 13 | eqtr4d 2796 | . 2 ⊢ (𝜑 → 𝑌 = ∪ 𝐾) |
15 | 4, 14 | sseqtrrd 3935 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3860 ∪ cuni 4801 ‘cfv 6340 (class class class)co 7156 ↾t crest 16765 Topctop 21606 Clsdccld 21729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-en 8541 df-fin 8544 df-fi 8921 df-rest 16767 df-topgen 16788 df-top 21607 df-topon 21624 df-bases 21659 df-cld 21732 |
This theorem is referenced by: restcls2 45646 restclssep 45648 iscnrm3llem1 45682 iscnrm3llem2 45683 |
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