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Mirrors > Home > MPE Home > Th. List > distop | Structured version Visualization version GIF version |
Description: The discrete topology on a set 𝐴. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 17-Jul-2006.) (Revised by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
distop | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniss 4849 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ ∪ 𝒫 𝐴) | |
2 | unipw 5346 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 1, 2 | sseqtrdi 4020 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ 𝐴) |
4 | vuniex 7468 | . . . . . 6 ⊢ ∪ 𝑥 ∈ V | |
5 | 4 | elpw 4546 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
6 | 3, 5 | sylibr 236 | . . . 4 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
7 | 6 | ax-gen 1795 | . . 3 ⊢ ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
8 | 7 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴)) |
9 | velpw 4547 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
10 | velpw 4547 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
11 | ssinss1 4217 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴) | |
12 | 11 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴)) |
13 | vex 3500 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
14 | 13 | inex2 5225 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ∈ V |
15 | 14 | elpw 4546 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐴) |
16 | 12, 15 | syl6ibr 254 | . . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
17 | 10, 16 | sylbi 219 | . . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
18 | 17 | com12 32 | . . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
19 | 9, 18 | sylbi 219 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
20 | 19 | ralrimiv 3184 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴) |
21 | 20 | rgen 3151 | . . 3 ⊢ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 |
22 | 21 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴) |
23 | pwexg 5282 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
24 | istopg 21506 | . . 3 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))) | |
25 | 23, 24 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))) |
26 | 8, 22, 25 | mpbir2and 711 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ∩ cin 3938 ⊆ wss 3939 𝒫 cpw 4542 ∪ cuni 4841 Topctop 21504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-pw 4544 df-sn 4571 df-pr 4573 df-uni 4842 df-top 21505 |
This theorem is referenced by: topnex 21607 distopon 21608 distps 21626 discld 21700 restdis 21789 dishaus 21993 discmp 22009 dis2ndc 22071 dislly 22108 dis1stc 22110 dissnlocfin 22140 locfindis 22141 txdis 22243 xkopt 22266 xkofvcn 22295 efmndtmd 22712 symgtgp 22717 dispcmp 31127 |
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