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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 4458 | . . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ ∪ 𝒫 𝐴) | |
2 | unipw 4918 | . . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 1, 2 | syl6sseq 3651 | . . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ 𝐴) |
4 | vuniex 6954 | . . . . . 6 ⊢ ∪ 𝑥 ∈ V | |
5 | 4 | elpw 4164 | . . . . 5 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴) |
6 | 3, 5 | sylibr 224 | . . . 4 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
7 | 6 | ax-gen 1722 | . . 3 ⊢ ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴) |
8 | 7 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴)) |
9 | selpw 4165 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
10 | selpw 4165 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴) | |
11 | ssinss1 3841 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴) | |
12 | 11 | a1i 11 | . . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴)) |
13 | vex 3203 | . . . . . . . . . . 11 ⊢ 𝑦 ∈ V | |
14 | 13 | inex2 4800 | . . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ∈ V |
15 | 14 | elpw 4164 | . . . . . . . . 9 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐴) |
16 | 12, 15 | syl6ibr 242 | . . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
17 | 10, 16 | sylbi 207 | . . . . . . 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 207 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)) |
20 | 19 | ralrimiv 2965 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴) |
21 | 20 | rgen 2922 | . . 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 4850 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V) | |
24 | istopg 20700 | . . 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 957 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 Topctop 20698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 df-uni 4437 df-top 20699 |
This theorem is referenced by: topnex 20800 distopon 20801 distps 20819 discld 20893 restdis 20982 dishaus 21186 discmp 21201 dis2ndc 21263 dislly 21300 dis1stc 21302 dissnlocfin 21332 locfindis 21333 txdis 21435 xkopt 21458 xkofvcn 21487 symgtgp 21905 dispcmp 29926 |
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