Theorem distop 12254

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 3757	. . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ ∪ 𝒫 𝐴)
2		unipw 4139	. . . . . 6 ⊢ ∪ 𝒫 𝐴 = 𝐴
3	1, 2	sseqtrdi 3145	. . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ⊆ 𝐴)
4		vuniex 4360	. . . . . 6 ⊢ ∪ 𝑥 ∈ V
5	4	elpw 3516	. . . . 5 ⊢ (∪ 𝑥 ∈ 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
6	3, 5	sylibr 133	. . . 4 ⊢ (𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴)
7	6	ax-gen 1425	. . 3 ⊢ ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴)
8	7	a1i 9	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥(𝑥 ⊆ 𝒫 𝐴 → ∪ 𝑥 ∈ 𝒫 𝐴))
9		velpw 3517	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
10		velpw 3517	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
11		ssinss1 3305	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴)
12	11	a1i 9	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ⊆ 𝐴))
13		vex 2689	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
14	13	inex2 4063	. . . . . . . . . 10 ⊢ (𝑥 ∩ 𝑦) ∈ V
15	14	elpw 3516	. . . . . . . . 9 ⊢ ((𝑥 ∩ 𝑦) ∈ 𝒫 𝐴 ↔ (𝑥 ∩ 𝑦) ⊆ 𝐴)
16	12, 15	syl6ibr 161	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
17	10, 16	sylbi 120	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → (𝑥 ⊆ 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
18	17	com12 30	. . . . . 6 ⊢ (𝑥 ⊆ 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
19	9, 18	sylbi 120	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → (𝑦 ∈ 𝒫 𝐴 → (𝑥 ∩ 𝑦) ∈ 𝒫 𝐴))
20	19	ralrimiv 2504	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 → ∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)
21	20	rgen 2485	. . 3 ⊢ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴
22	21	a1i 9	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝐴(𝑥 ∩ 𝑦) ∈ 𝒫 𝐴)
23		pwexg 4104	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
24		istopg 12166	. . 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 14	. 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 928	1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ∩ cin 3070   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736  Topctop 12164

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-uni 3737  df-top 12165

This theorem is referenced by:  topnex  12255  distopon  12256  distps  12260  discld  12305  restdis  12353  txdis  12446