Theorem ufildr 24056

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 4908	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
2		ufilfil 24029	. . . . . . . . 9 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3		filunibas 24006	. . . . . . . . 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 4888	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ (𝐹 ∪ {∅})
7		uniun 4899	. . . . . . . . . 10 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
8		0ex 5272	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
9	8	unisn 4895	. . . . . . . . . . 11 ⊢ ∪ {∅} = ∅
10	9	uneq2i 4127	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
11		un0 4358	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
12	7, 10, 11	3eqtri 2796	. . . . . . . . 9 ⊢ ∪ (𝐹 ∪ {∅}) = ∪ 𝐹
13	6, 12	eqtr2i 2793	. . . . . . . 8 ⊢ ∪ 𝐹 = ∪ 𝐽
14	4, 13	eqtr3di 2819	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	sseq2d 3977	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 ↔ 𝑥 ⊆ ∪ 𝐽))
16	1, 15	imbitrrid 249	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
17		eqid 2769	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	cldss 23154	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
19	18, 15	imbitrrid 249	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋))
20	16, 19	jaod 872	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑋))
21		ufilss 24030	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))
22		ssun1 4139	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
23	22, 5	sseqtrri 3994	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐽
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐽)
25	24	sseld 3944	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝐽))
26	24	sseld 3944	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐽))
27		filconn 24008	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28		conntop 23542	. . . . . . . . . . . 12 ⊢ ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
29	2, 27, 28	3syl 19	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
30	5, 29	eqeltrid 2873	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
31	15	biimpa 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ∪ 𝐽)
32	17	iscld2 23153	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
33	30, 31, 32	syl2an2r 697	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
34	14	difeq1d 4088	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑋 ∖ 𝑥) = (∪ 𝐽 ∖ 𝑥))
35	34	eleq1d 2854	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
36	35	adantr 485	. . . . . . . . 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 417	. . . 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 4115	. . 3 ⊢ (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
44		velpw 4572	. . 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 2767	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  ∪ cuni 4876  ‘cfv 6537  Topctop 23018  Clsdccld 23141  Conncconn 23536  Filcfil 23970  UFilcufil 24024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-fbas 21487  df-top 23019  df-cld 23144  df-conn 23537  df-fil 23971  df-ufil 24026

This theorem is referenced by: (None)