Theorem ufildr 22541

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 4870	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
2		ufildr.1	. . . . . . . . . 10 ⊢ 𝐽 = (𝐹 ∪ {∅})
3	2	unieqi 4853	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ (𝐹 ∪ {∅})
4		uniun 4863	. . . . . . . . . 10 ⊢ ∪ (𝐹 ∪ {∅}) = (∪ 𝐹 ∪ ∪ {∅})
5		0ex 5213	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
6	5	unisn 4860	. . . . . . . . . . 11 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4138	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∪ {∅}) = (∪ 𝐹 ∪ ∅)
8		un0 4346	. . . . . . . . . 10 ⊢ (∪ 𝐹 ∪ ∅) = ∪ 𝐹
9	4, 7, 8	3eqtri 2850	. . . . . . . . 9 ⊢ ∪ (𝐹 ∪ {∅}) = ∪ 𝐹
10	3, 9	eqtr2i 2847	. . . . . . . 8 ⊢ ∪ 𝐹 = ∪ 𝐽
11		ufilfil 22514	. . . . . . . . 9 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
12		filunibas 22491	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (UFil‘𝑋) → ∪ 𝐹 = 𝑋)
14	10, 13	syl5reqr 2873	. . . . . . 7 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝑋 = ∪ 𝐽)
15	14	sseq2d 4001	. . . . . 6 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 ↔ 𝑥 ⊆ ∪ 𝐽))
16	1, 15	syl5ibr 248	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
17		eqid 2823	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
18	17	cldss 21639	. . . . . 6 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ ∪ 𝐽)
19	18, 15	syl5ibr 248	. . . . 5 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋))
20	16, 19	jaod 855	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) → 𝑥 ⊆ 𝑋))
21		ufilss 22515	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹))
22		ssun1 4150	. . . . . . . . . 10 ⊢ 𝐹 ⊆ (𝐹 ∪ {∅})
23	22, 2	sseqtrri 4006	. . . . . . . . 9 ⊢ 𝐹 ⊆ 𝐽
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝐹 ⊆ 𝐽)
25	24	sseld 3968	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐹 → 𝑥 ∈ 𝐽))
26	24	sseld 3968	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → (𝑋 ∖ 𝑥) ∈ 𝐽))
27		filconn 22493	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28		conntop 22027	. . . . . . . . . . . 12 ⊢ ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
29	11, 27, 28	3syl 18	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
30	2, 29	eqeltrid 2919	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
31	15	biimpa 479	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ⊆ ∪ 𝐽)
32	17	iscld2 21638	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ ∪ 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
33	30, 31, 32	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
34	14	difeq1d 4100	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑋 ∖ 𝑥) = (∪ 𝐽 ∖ 𝑥))
35	34	eleq1d 2899	. . . . . . . . . 10 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
36	35	adantr 483	. . . . . . . . 9 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐽 ↔ (∪ 𝐽 ∖ 𝑥) ∈ 𝐽))
37	33, 36	bitr4d 284	. . . . . . . 8 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ 𝐽))
38	26, 37	sylibrd 261	. . . . . . 7 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑋 ∖ 𝑥) ∈ 𝐹 → 𝑥 ∈ (Clsd‘𝐽)))
39	25, 38	orim12d 961	. . . . . 6 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
40	21, 39	mpd 15	. . . . 5 ⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
41	40	ex 415	. . . 4 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ⊆ 𝑋 → (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽))))
42	20, 41	impbid 214	. . 3 ⊢ (𝐹 ∈ (UFil‘𝑋) → ((𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥 ⊆ 𝑋))
43		elun 4127	. . 3 ⊢ (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥 ∈ 𝐽 ∨ 𝑥 ∈ (Clsd‘𝐽)))
44		velpw 4546	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
45	42, 43, 44	3bitr4g 316	. 2 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
46	45	eqrdv 2821	1 ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ∖ cdif 3935   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {csn 4569  ∪ cuni 4840  ‘cfv 6357  Topctop 21503  Clsdccld 21626  Conncconn 22021  Filcfil 22455  UFilcufil 22509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365  df-fbas 20544  df-top 21504  df-cld 21629  df-conn 22022  df-fil 22456  df-ufil 22511

This theorem is referenced by: (None)