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Theorem ufildr 23955
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 4942 . . . . . 6 (𝑥𝐽𝑥 𝐽)
2 ufilfil 23928 . . . . . . . . 9 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filunibas 23905 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
42, 3syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
5 ufildr.1 . . . . . . . . . 10 𝐽 = (𝐹 ∪ {∅})
65unieqi 4924 . . . . . . . . 9 𝐽 = (𝐹 ∪ {∅})
7 uniun 4935 . . . . . . . . . 10 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
8 0ex 5313 . . . . . . . . . . . 12 ∅ ∈ V
98unisn 4931 . . . . . . . . . . 11 {∅} = ∅
109uneq2i 4175 . . . . . . . . . 10 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
11 un0 4400 . . . . . . . . . 10 ( 𝐹 ∪ ∅) = 𝐹
127, 10, 113eqtri 2767 . . . . . . . . 9 (𝐹 ∪ {∅}) = 𝐹
136, 12eqtr2i 2764 . . . . . . . 8 𝐹 = 𝐽
144, 13eqtr3di 2790 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋 = 𝐽)
1514sseq2d 4028 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋𝑥 𝐽))
161, 15imbitrrid 246 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝐽𝑥𝑋))
17 eqid 2735 . . . . . . 7 𝐽 = 𝐽
1817cldss 23053 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
1918, 15imbitrrid 246 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋))
2016, 19jaod 859 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝑋))
21 ufilss 23929 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
22 ssun1 4188 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {∅})
2322, 5sseqtrri 4033 . . . . . . . . 9 𝐹𝐽
2423a1i 11 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹𝐽)
2524sseld 3994 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹𝑥𝐽))
2624sseld 3994 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐽))
27 filconn 23907 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28 conntop 23441 . . . . . . . . . . . 12 ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
292, 27, 283syl 18 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
305, 29eqeltrid 2843 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
3115biimpa 476 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝑥 𝐽)
3217iscld2 23052 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3330, 31, 32syl2an2r 685 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3414difeq1d 4135 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝑋𝑥) = ( 𝐽𝑥))
3534eleq1d 2824 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3635adantr 480 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3733, 36bitr4d 282 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ 𝐽))
3826, 37sylibrd 259 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹𝑥 ∈ (Clsd‘𝐽)))
3925, 38orim12d 966 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4021, 39mpd 15 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
4140ex 412 . . . 4 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋 → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4220, 41impbid 212 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥𝑋))
43 elun 4163 . . 3 (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
44 velpw 4610 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
4542, 43, 443bitr4g 314 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
4645eqrdv 2733 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  cdif 3960  cun 3961  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  Conncconn 23435  Filcfil 23869  UFilcufil 23923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-fbas 21379  df-top 22916  df-cld 23043  df-conn 23436  df-fil 23870  df-ufil 23925
This theorem is referenced by: (None)
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