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Mirrors > Home > MPE Home > Th. List > Mathboxes > restcls2 | Structured version Visualization version GIF version |
Description: A closed set in a subspace topology is the closure in the original topology intersecting with 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 |
---|---|
restcls2 | ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
2 | 1 | fveq2d 6889 | . . 3 ⊢ (𝜑 → (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌))) |
3 | 2 | fveq1d 6887 | . 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆)) |
4 | restcls2.5 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | |
5 | cldcls 22901 | . . 3 ⊢ (𝑆 ∈ (Clsd‘𝐾) → ((cls‘𝐾)‘𝑆) = 𝑆) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = 𝑆) |
7 | restcls2.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
8 | restcls2.3 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
9 | restcls2.2 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | |
10 | 8, 9 | sseqtrd 4017 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
11 | 7, 9, 8, 1, 4 | restcls2lem 47816 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
12 | eqid 2726 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | eqid 2726 | . . . 4 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
14 | 12, 13 | restcls 23040 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
15 | 7, 10, 11, 14 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
16 | 3, 6, 15 | 3eqtr3d 2774 | 1 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ cin 3942 ⊆ wss 3943 ∪ cuni 4902 ‘cfv 6537 (class class class)co 7405 ↾t crest 17375 Topctop 22750 Clsdccld 22875 clsccl 22877 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-en 8942 df-fin 8945 df-fi 9408 df-rest 17377 df-topgen 17398 df-top 22751 df-topon 22768 df-bases 22804 df-cld 22878 df-cls 22880 |
This theorem is referenced by: restclsseplem 47818 |
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