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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 4394 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ ∪ 𝒫 𝐴) | |
2 | unipw 4845 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 1, 2 | syl6sseq 3614 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ 𝐴) |
4 | vuniex 6852 | . . . . . 6 ⊢ ∪ 𝑥 ∈ V | |
5 | 4 | elpw 4114 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
6 | 3, 5 | sylibr 223 | . . . 4 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
7 | 6 | ax-gen 1713 | . . 3 ⊢ ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
8 | 7 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴)) |
9 | selpw 4115 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
10 | selpw 4115 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
11 | ssinss1 3803 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴) | |
12 | 11 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴)) |
13 | vex 3176 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
14 | 13 | inex2 4728 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ∈ V |
15 | 14 | elpw 4114 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐴) |
16 | 12, 15 | syl6ibr 241 | . . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
17 | 10, 16 | sylbi 206 | . . . . . . 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 206 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
20 | 19 | ralrimiv 2948 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴) |
21 | 20 | rgen 2906 | . . 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 4776 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
24 | istopg 20525 | . . 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 959 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 Topctop 20517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-sn 4126 df-pr 4128 df-uni 4373 df-top 20521 |
This theorem is referenced by: distopon 20611 distps 20629 discld 20703 restdis 20792 dishaus 20996 discmp 21011 dis2ndc 21073 dislly 21110 dis1stc 21112 dissnlocfin 21142 locfindis 21143 txdis 21245 xkopt 21268 xkofvcn 21297 symgtgp 21715 dispcmp 29254 bj-topnex 32247 |
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