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Theorem ufildr 22028
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 4627 . . . . . 6 (𝑥𝐽𝑥 𝐽)
2 ufildr.1 . . . . . . . . . 10 𝐽 = (𝐹 ∪ {∅})
32unieqi 4605 . . . . . . . . 9 𝐽 = (𝐹 ∪ {∅})
4 uniun 4617 . . . . . . . . . 10 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
5 0ex 4952 . . . . . . . . . . . 12 ∅ ∈ V
65unisn 4612 . . . . . . . . . . 11 {∅} = ∅
76uneq2i 3928 . . . . . . . . . 10 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
8 un0 4131 . . . . . . . . . 10 ( 𝐹 ∪ ∅) = 𝐹
94, 7, 83eqtri 2791 . . . . . . . . 9 (𝐹 ∪ {∅}) = 𝐹
103, 9eqtr2i 2788 . . . . . . . 8 𝐹 = 𝐽
11 ufilfil 22001 . . . . . . . . 9 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
12 filunibas 21978 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1311, 12syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
1410, 13syl5reqr 2814 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋 = 𝐽)
1514sseq2d 3795 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋𝑥 𝐽))
161, 15syl5ibr 237 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝐽𝑥𝑋))
17 eqid 2765 . . . . . . 7 𝐽 = 𝐽
1817cldss 21127 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
1918, 15syl5ibr 237 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋))
2016, 19jaod 885 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝑋))
21 ufilss 22002 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
22 ssun1 3940 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {∅})
2322, 2sseqtr4i 3800 . . . . . . . . 9 𝐹𝐽
2423a1i 11 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹𝐽)
2524sseld 3762 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹𝑥𝐽))
2624sseld 3762 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐽))
27 filconn 21980 . . . . . . . . . . . . 13 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)
28 conntop 21514 . . . . . . . . . . . . 13 ((𝐹 ∪ {∅}) ∈ Conn → (𝐹 ∪ {∅}) ∈ Top)
2911, 27, 283syl 18 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
302, 29syl5eqel 2848 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
3130adantr 472 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
3215biimpa 468 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝑥 𝐽)
3317iscld2 21126 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3431, 32, 33syl2anc 579 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3514difeq1d 3891 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝑋𝑥) = ( 𝐽𝑥))
3635eleq1d 2829 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3736adantr 472 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3834, 37bitr4d 273 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ 𝐽))
3926, 38sylibrd 250 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹𝑥 ∈ (Clsd‘𝐽)))
4025, 39orim12d 987 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4121, 40mpd 15 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
4241ex 401 . . . 4 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋 → (𝑥𝐽𝑥 ∈ (Clsd‘𝐽))))
4320, 42impbid 203 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥𝑋))
44 elun 3917 . . 3 (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
45 selpw 4324 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
4643, 44, 453bitr4g 305 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
4746eqrdv 2763 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  cdif 3731  cun 3732  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336   cuni 4596  cfv 6070  Topctop 20991  Clsdccld 21114  Conncconn 21508  Filcfil 21942  UFilcufil 21996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-fv 6078  df-fbas 20030  df-top 20992  df-cld 21117  df-conn 21509  df-fil 21943  df-ufil 21998
This theorem is referenced by: (None)
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