MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ufildr Structured version   Visualization version   GIF version

Theorem ufildr 21928
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.
StepHypRef Expression
1 elssuni 4611 . . . . . 6 (𝑥𝐽𝑥 𝐽)
2 ufildr.1 . . . . . . . . . 10 𝐽 = (𝐹 ∪ {∅})
32unieqi 4589 . . . . . . . . 9 𝐽 = (𝐹 ∪ {∅})
4 uniun 4600 . . . . . . . . . 10 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
5 0ex 4934 . . . . . . . . . . . 12 ∅ ∈ V
65unisn 4595 . . . . . . . . . . 11 {∅} = ∅
76uneq2i 3899 . . . . . . . . . 10 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
8 un0 4102 . . . . . . . . . 10 ( 𝐹 ∪ ∅) = 𝐹
94, 7, 83eqtri 2778 . . . . . . . . 9 (𝐹 ∪ {∅}) = 𝐹
103, 9eqtr2i 2775 . . . . . . . 8 𝐹 = 𝐽
11 ufilfil 21901 . . . . . . . . 9 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
12 filunibas 21878 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1311, 12syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
1410, 13syl5reqr 2801 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋 = 𝐽)
1514sseq2d 3766 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋𝑥 𝐽))
161, 15syl5ibr 236 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝐽𝑥𝑋))
17 eqid 2752 . . . . . . 7 𝐽 = 𝐽
1817cldss 21027 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
1918, 15syl5ibr 236 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋))
2016, 19jaod 394 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝑋))
21 ufilss 21902 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
22 ssun1 3911 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {∅})
2322, 2sseqtr4i 3771 . . . . . . . . 9 𝐹𝐽
2423a1i 11 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹𝐽)
2524sseld 3735 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹𝑥𝐽))
2624sseld 3735 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐽))
27 filconn 21880 . . . . . . . . . . . . 13 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28 conntop 21414 . . . . . . . . . . . . 13 ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
2911, 27, 283syl 18 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
302, 29syl5eqel 2835 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
3130adantr 472 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
3215biimpa 502 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝑥 𝐽)
3317iscld2 21026 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3431, 32, 33syl2anc 696 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3514difeq1d 3862 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝑋𝑥) = ( 𝐽𝑥))
3635eleq1d 2816 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3736adantr 472 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3834, 37bitr4d 271 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ 𝐽))
3926, 38sylibrd 249 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹𝑥 ∈ (Clsd‘𝐽)))
4025, 39orim12d 919 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4121, 40mpd 15 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
4241ex 449 . . . 4 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋 → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4320, 42impbid 202 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥𝑋))
44 elun 3888 . . 3 (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
45 selpw 4301 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
4643, 44, 453bitr4g 303 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
4746eqrdv 2750 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1624  wcel 2131  cdif 3704  cun 3705  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   cuni 4580  cfv 6041  Topctop 20892  Clsdccld 21014  Conncconn 21408  Filcfil 21842  UFilcufil 21896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-fv 6049  df-fbas 19937  df-top 20893  df-cld 21017  df-conn 21409  df-fil 21843  df-ufil 21898
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator