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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 2727 | . . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
3 | 2 | cldss 22920 | . . 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 4018 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
9 | eqid 2727 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
10 | 9 | restuni 23053 | . . . 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 4916 | . . 3 ⊢ (𝜑 → ∪ 𝐾 = ∪ (𝐽 ↾t 𝑌)) |
14 | 11, 13 | eqtr4d 2770 | . 2 ⊢ (𝜑 → 𝑌 = ∪ 𝐾) |
15 | 4, 14 | sseqtrrd 4019 | 1 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ∪ cuni 4903 ‘cfv 6542 (class class class)co 7414 ↾t crest 17393 Topctop 22782 Clsdccld 22907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-en 8956 df-fin 8959 df-fi 9426 df-rest 17395 df-topgen 17416 df-top 22783 df-topon 22800 df-bases 22836 df-cld 22910 |
This theorem is referenced by: restcls2 47855 restclssep 47857 iscnrm3llem1 47891 iscnrm3llem2 47892 |
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