Theorem distop 22880

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.)

Assertion

Ref	Expression
distop	⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)

Proof of Theorem distop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4866	. . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ ∪ 𝒫 𝐴)
2		unipw 5393	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	1, 2	sseqtrdi 3976	. . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ 𝐴)
4		vuniex 7675	. . . . . 6 ⊢ ∪ 𝑥 ∈ V
5	4	elpw 4555	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
6	3, 5	sylibr 234	. . . 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 4556	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
10		velpw 4556	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
11		ssinss1 4197	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴)
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴))
13		vex 3440	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
14	13	inex2 5257	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	elpw 4555	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐴)
16	12, 15	imbitrrdi 252	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
17	10, 16	sylbi 217	. . . . . . 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 217	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
20	19	ralrimiv 3120	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)
21	20	rgen 3046	. . 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 5317	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
24		istopg 22780	. . 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 713	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  Topctop 22778

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-un 3908  df-in 3910  df-ss 3920  df-pw 4553  df-sn 4578  df-pr 4580  df-uni 4859  df-top 22779

This theorem is referenced by:  topnex  22881  distopon  22882  distps  22900  discld  22974  restdis  23063  dishaus  23267  discmp  23283  dis2ndc  23345  dislly  23382  dis1stc  23384  dissnlocfin  23414  locfindis  23415  txdis  23517  xkopt  23540  xkofvcn  23569  efmndtmd  23986  symgtgp  23991  dispcmp  33842