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Mirrors > Home > ILE Home > Th. List > isopn3 | 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 12206 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) |
3 | inss2 3267 | . . . . . . . 8 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝒫 𝑆 | |
4 | 3 | unissi 3729 | . . . . . . 7 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ ∪ 𝒫 𝑆 |
5 | unipw 4109 | . . . . . . 7 ⊢ ∪ 𝒫 𝑆 = 𝑆 | |
6 | 4, 5 | sseqtri 3101 | . . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆 |
7 | 6 | a1i 9 | . . . . 5 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝑆) |
8 | id 19 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝐽) | |
9 | pwidg 3494 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ 𝒫 𝑆) | |
10 | 8, 9 | elind 3231 | . . . . . 6 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ∈ (𝐽 ∩ 𝒫 𝑆)) |
11 | elssuni 3734 | . . . . . 6 ⊢ (𝑆 ∈ (𝐽 ∩ 𝒫 𝑆) → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆)) | |
12 | 10, 11 | syl 14 | . . . . 5 ⊢ (𝑆 ∈ 𝐽 → 𝑆 ⊆ ∪ (𝐽 ∩ 𝒫 𝑆)) |
13 | 7, 12 | eqssd 3084 | . . . 4 ⊢ (𝑆 ∈ 𝐽 → ∪ (𝐽 ∩ 𝒫 𝑆) = 𝑆) |
14 | 2, 13 | sylan9eq 2170 | . . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑆 ∈ 𝐽) → ((int‘𝐽)‘𝑆) = 𝑆) |
15 | 14 | ex 114 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 → ((int‘𝐽)‘𝑆) = 𝑆)) |
16 | 1 | ntropn 12213 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) |
17 | eleq1 2180 | . . 3 ⊢ (((int‘𝐽)‘𝑆) = 𝑆 → (((int‘𝐽)‘𝑆) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽)) | |
18 | 16, 17 | syl5ibcom 154 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (((int‘𝐽)‘𝑆) = 𝑆 → 𝑆 ∈ 𝐽)) |
19 | 15, 18 | impbid 128 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 ∩ cin 3040 ⊆ wss 3041 𝒫 cpw 3480 ∪ cuni 3706 ‘cfv 5093 Topctop 12091 intcnt 12189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-top 12092 df-ntr 12192 |
This theorem is referenced by: ntridm 12222 ntrtop 12224 ntr0 12230 isopn3i 12231 cnntr 12321 |
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