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Theorem ufildr 23828
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 4935 . . . . . 6 (𝑥𝐽𝑥 𝐽)
2 ufilfil 23801 . . . . . . . . 9 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filunibas 23778 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
42, 3syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
5 ufildr.1 . . . . . . . . . 10 𝐽 = (𝐹 ∪ {∅})
65unieqi 4915 . . . . . . . . 9 𝐽 = (𝐹 ∪ {∅})
7 uniun 4928 . . . . . . . . . 10 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
8 0ex 5301 . . . . . . . . . . . 12 ∅ ∈ V
98unisn 4924 . . . . . . . . . . 11 {∅} = ∅
109uneq2i 4156 . . . . . . . . . 10 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
11 un0 4386 . . . . . . . . . 10 ( 𝐹 ∪ ∅) = 𝐹
127, 10, 113eqtri 2760 . . . . . . . . 9 (𝐹 ∪ {∅}) = 𝐹
136, 12eqtr2i 2757 . . . . . . . 8 𝐹 = 𝐽
144, 13eqtr3di 2783 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋 = 𝐽)
1514sseq2d 4010 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋𝑥 𝐽))
161, 15imbitrrid 245 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝐽𝑥𝑋))
17 eqid 2728 . . . . . . 7 𝐽 = 𝐽
1817cldss 22926 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
1918, 15imbitrrid 245 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋))
2016, 19jaod 858 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝑋))
21 ufilss 23802 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
22 ssun1 4168 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {∅})
2322, 5sseqtrri 4015 . . . . . . . . 9 𝐹𝐽
2423a1i 11 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹𝐽)
2524sseld 3977 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹𝑥𝐽))
2624sseld 3977 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐽))
27 filconn 23780 . . . . . . . . . . . 12 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28 conntop 23314 . . . . . . . . . . . 12 ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
292, 27, 283syl 18 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
305, 29eqeltrid 2833 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
3115biimpa 476 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝑥 𝐽)
3217iscld2 22925 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3330, 31, 32syl2an2r 684 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3414difeq1d 4117 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝑋𝑥) = ( 𝐽𝑥))
3534eleq1d 2814 . . . . . . . . . 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 963 . . . . . 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 211 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥𝑋))
43 elun 4144 . . 3 (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
44 velpw 4603 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
4542, 43, 443bitr4g 314 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
4645eqrdv 2726 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  cdif 3942  cun 3943  wss 3945  c0 4318  𝒫 cpw 4598  {csn 4624   cuni 4903  cfv 6542  Topctop 22788  Clsdccld 22913  Conncconn 23308  Filcfil 23742  UFilcufil 23796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-fbas 21269  df-top 22789  df-cld 22916  df-conn 23309  df-fil 23743  df-ufil 23798
This theorem is referenced by: (None)
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