Theorem ufildr 24139

Description: An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)

Hypothesis

Ref	Expression
ufildr.1	⊢ 𝐽 = (𝐹 ∪ {∅})

Assertion

Ref	Expression
ufildr	⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

Proof of Theorem ufildr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elssuni 4906	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
2		ufilfil 24112	. . . . . . . . 9 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filunibas 24089	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
5		ufildr.1	. . . . . . . . . 10 ⊢ 𝐽 = (𝐹 ∪ {∅})
6	5	unieqi 4886	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ (𝐹 ∪ {∅})
7		uniun 4897	. . . . . . . . . 10 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
8		0ex 5272	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
9	8	unisn 4893	. . . . . . . . . . 11 ⊢ ∪ {∅} = ∅
10	9	uneq2i 4119	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
11		un0 4351	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
12	7, 10, 11	3eqtri 2792	. . . . . . . . 9 ⊢ ∪ (𝐹 ∪ {∅}) = ∪ 𝐹
13	6, 12	eqtr2i 2789	. . . . . . . 8 ⊢ ∪ 𝐹 = ∪ 𝐽
14	4, 13	eqtr3di 2815	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	sseq2d 3970	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 ↔ 𝑥 ⊆ ∪ 𝐽))
16	1, 15	imbitrrid 249	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
17		eqid 2765	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	cldss 23236	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
19	18, 15	imbitrrid 249	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋))
20	16, 19	jaod 873	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑋))
21		ufilss 24113	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))
22		ssun1 4131	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
23	22, 5	sseqtrri 3987	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐽
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐽)
25	24	sseld 3937	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝐽))
26	24	sseld 3937	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐽))
27		filconn 24091	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28		conntop 23624	. . . . . . . . . . . 12 ⊢ ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
29	2, 27, 28	3syl 19	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
30	5, 29	eqeltrid 2869	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
31	15	biimpa 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ∪ 𝐽)
32	17	iscld2 23235	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
33	30, 31, 32	syl2an2r 698	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
34	14	difeq1d 4080	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑋 ∖ 𝑥) = (∪ 𝐽 ∖ 𝑥))
35	34	eleq1d 2850	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
36	35	adantr 486	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
37	33, 36	bitr4d 285	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
38	26, 37	sylibrd 262	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → 𝑥 ∈ (Clsd‘𝐽)))
39	25, 38	orim12d 979	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
40	21, 39	mpd 16	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
41	40	ex 418	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
42	20, 41	impbid 215	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥 ⊆ 𝑋))
43		elun 4107	. . 3 ⊢ (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
44		velpw 4569	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
45	42, 43, 44	3bitr4g 317	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
46	45	eqrdv 2763	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874  ‘cfv 6540  Topctop 23100  Clsdccld 23223  Conncconn 23618  Filcfil 24053  UFilcufil 24107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-fbas 21569  df-top 23101  df-cld 23226  df-conn 23619  df-fil 24054  df-ufil 24109

This theorem is used by: (None)