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Mirrors > Home > MPE Home > Th. List > Mathboxes > restclssep | Structured version Visualization version GIF version |
Description: Two disjoint closed sets in a subspace topology are separated in the original topology. (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)) |
restclsseplem.6 | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
restclssep.7 | ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾)) |
Ref | Expression |
---|---|
restclssep | ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4160 | . . 3 ⊢ (((cls‘𝐽)‘𝑇) ∩ 𝑆) = (𝑆 ∩ ((cls‘𝐽)‘𝑇)) | |
2 | restcls2.1 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | restcls2.2 | . . . 4 ⊢ (𝜑 → 𝑋 = ∪ 𝐽) | |
4 | restcls2.3 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
5 | restcls2.4 | . . . 4 ⊢ (𝜑 → 𝐾 = (𝐽 ↾t 𝑌)) | |
6 | restclssep.7 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐾)) | |
7 | incom 4160 | . . . . 5 ⊢ (𝑆 ∩ 𝑇) = (𝑇 ∩ 𝑆) | |
8 | restclsseplem.6 | . . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) | |
9 | 7, 8 | eqtr3id 2792 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅) |
10 | restcls2.5 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐾)) | |
11 | 2, 3, 4, 5, 10 | restcls2lem 46700 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑌) |
12 | 2, 3, 4, 5, 6, 9, 11 | restclsseplem 46702 | . . 3 ⊢ (𝜑 → (((cls‘𝐽)‘𝑇) ∩ 𝑆) = ∅) |
13 | 1, 12 | eqtr3id 2792 | . 2 ⊢ (𝜑 → (𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅) |
14 | 2, 3, 4, 5, 6 | restcls2lem 46700 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑌) |
15 | 2, 3, 4, 5, 10, 8, 14 | restclsseplem 46702 | . 2 ⊢ (𝜑 → (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅) |
16 | 13, 15 | jca 513 | 1 ⊢ (𝜑 → ((𝑆 ∩ ((cls‘𝐽)‘𝑇)) = ∅ ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑇) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∩ cin 3908 ⊆ wss 3909 ∅c0 4281 ∪ cuni 4864 ‘cfv 6492 (class class class)co 7350 ↾t crest 17237 Topctop 22165 Clsdccld 22290 clsccl 22292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-en 8818 df-fin 8821 df-fi 9281 df-rest 17239 df-topgen 17260 df-top 22166 df-topon 22183 df-bases 22219 df-cld 22293 df-cls 22295 |
This theorem is referenced by: iscnrm3l 46739 |
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