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Mirrors > Home > MPE Home > Th. List > isopn3 | Structured version Visualization version GIF version |
Description: A subset is open iff it equals its own interior. (Contributed by NM, 9-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
isopn3 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clscld.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | ntrval 22933 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) |
3 | inss2 4225 | . . . . . . . 8 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆 | |
4 | 3 | unissi 4912 | . . . . . . 7 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆 |
5 | unipw 5446 | . . . . . . 7 ⊢ ∪ 𝒫 𝑆 = 𝑆 | |
6 | 4, 5 | sseqtri 4014 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆 |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆) |
8 | id 22 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝐽) | |
9 | pwidg 4618 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝒫 𝑆) | |
10 | 8, 9 | elind 4190 | . . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ (𝐽 ∩ 𝒫 𝑆)) |
11 | elssuni 4935 | . . . . . 6 ⊢ (𝑆 ∈ (𝐽 ∩ 𝒫 𝑆) → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆)) | |
12 | 10, 11 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆)) |
13 | 7, 12 | eqssd 3995 | . . . 4 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) = 𝑆) |
14 | 2, 13 | sylan9eq 2787 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
15 | 14 | ex 412 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 → ((int‘𝐽)‘𝑆) = 𝑆)) |
16 | 1 | ntropn 22946 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) |
17 | eleq1 2816 | . . 3 ⊢ (((int‘𝐽)‘𝑆) = 𝑆 → (((int‘𝐽)‘𝑆) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽)) | |
18 | 16, 17 | syl5ibcom 244 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = 𝑆 → 𝑆 ∈ 𝐽)) |
19 | 15, 18 | impbid 211 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4598 ∪ cuni 4903 ‘cfv 6542 Topctop 22788 intcnt 22914 |
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-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-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 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-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-top 22789 df-ntr 22917 |
This theorem is referenced by: ntridm 22965 ntrtop 22967 ntr0 22978 isopn3i 22979 opnnei 23017 cnntr 23172 llycmpkgen2 23447 dvnres 25854 dvcnvre 25945 taylthlem2 26302 taylthlem2OLD 26303 ulmdvlem3 26331 abelth 26371 opnbnd 35799 ioontr 44868 cncfuni 45246 fperdvper 45279 dirkercncflem3 45465 dirkercncflem4 45466 fourierdlem58 45524 fourierdlem73 45539 |
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