Theorem ufildr 21735

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 4467	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
2		ufildr.1	. . . . . . . . . 10 ⊢ 𝐽 = (𝐹 ∪ {∅})
3	2	unieqi 4445	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ (𝐹 ∪ {∅})
4		uniun 4456	. . . . . . . . . 10 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
5		0ex 4790	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	unisn 4451	. . . . . . . . . . 11 ⊢ ∪ {∅} = ∅
7	6	uneq2i 3764	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
8		un0 3967	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
9	4, 7, 8	3eqtri 2648	. . . . . . . . 9 ⊢ ∪ (𝐹 ∪ {∅}) = ∪ 𝐹
10	3, 9	eqtr2i 2645	. . . . . . . 8 ⊢ ∪ 𝐹 = ∪ 𝐽
11		ufilfil 21708	. . . . . . . . 9 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
12		filunibas 21685	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
14	10, 13	syl5reqr 2671	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	sseq2d 3633	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 ↔ 𝑥 ⊆ ∪ 𝐽))
16	1, 15	syl5ibr 236	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
17		eqid 2622	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	cldss 20833	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
19	18, 15	syl5ibr 236	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋))
20	16, 19	jaod 395	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑋))
21		ufilss 21709	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))
22		ssun1 3776	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
23	22, 2	sseqtr4i 3638	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐽
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐽)
25	24	sseld 3602	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝐽))
26	24	sseld 3602	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐽))
27		filconn 21687	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28		conntop 21220	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
29	11, 27, 28	3syl 18	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
30	2, 29	syl5eqel 2705	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
31	30	adantr 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐽 ∈ Top)
32	15	biimpa 501	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ∪ 𝐽)
33	17	iscld2 20832	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
34	31, 32, 33	syl2anc 693	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
35	14	difeq1d 3727	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑋 ∖ 𝑥) = (∪ 𝐽 ∖ 𝑥))
36	35	eleq1d 2686	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
37	36	adantr 481	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
38	34, 37	bitr4d 271	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
39	26, 38	sylibrd 249	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → 𝑥 ∈ (Clsd‘𝐽)))
40	25, 39	orim12d 883	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
41	21, 40	mpd 15	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
42	41	ex 450	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
43	20, 42	impbid 202	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥 ⊆ 𝑋))
44		elun 3753	. . 3 ⊢ (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
45		selpw 4165	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
46	43, 44, 45	3bitr4g 303	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
47	46	eqrdv 2620	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  Clsdccld 20820  Conncconn 21214  Filcfil 21649  UFilcufil 21703

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-fbas 19743  df-top 20699  df-cld 20823  df-conn 21215  df-fil 21650  df-ufil 21705

This theorem is referenced by: (None)