![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
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 6896 | . . 3 ⊢ (𝜑 → (cls‘𝐾) = (cls‘(𝐽 ↾t 𝑌))) |
3 | 2 | fveq1d 6894 | . 2 ⊢ (𝜑 → ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽 ↾t 𝑌))‘𝑆)) |
4 | restcls2.5 | . . 3 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | |
5 | cldcls 22940 | . . 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 4019 | . . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ 𝐽) |
11 | 7, 9, 8, 1, 4 | restcls2lem 47922 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
12 | eqid 2728 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
13 | eqid 2728 | . . . 4 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
14 | 12, 13 | restcls 23079 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ 𝑌) → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
15 | 7, 10, 11, 14 | syl3anc 1369 | . 2 ⊢ (𝜑 → ((cls‘(𝐽 ↾t 𝑌))‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
16 | 3, 6, 15 | 3eqtr3d 2776 | 1 ⊢ (𝜑 → 𝑆 = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ∪ cuni 4904 ‘cfv 6543 (class class class)co 7415 ↾t crest 17396 Topctop 22789 Clsdccld 22914 clsccl 22916 |
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 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-en 8959 df-fin 8962 df-fi 9429 df-rest 17398 df-topgen 17419 df-top 22790 df-topon 22807 df-bases 22843 df-cld 22917 df-cls 22919 |
This theorem is referenced by: restclsseplem 47924 |
Copyright terms: Public domain | W3C validator |