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Theorem ufildr 21545
 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 4403 . . . . . 6 (𝑥𝐽𝑥 𝐽)
2 ufildr.1 . . . . . . . . . 10 𝐽 = (𝐹 ∪ {∅})
32unieqi 4381 . . . . . . . . 9 𝐽 = (𝐹 ∪ {∅})
4 uniun 4392 . . . . . . . . . 10 (𝐹 ∪ {∅}) = ( 𝐹 {∅})
5 0ex 4718 . . . . . . . . . . . 12 ∅ ∈ V
65unisn 4387 . . . . . . . . . . 11 {∅} = ∅
76uneq2i 3726 . . . . . . . . . 10 ( 𝐹 {∅}) = ( 𝐹 ∪ ∅)
8 un0 3919 . . . . . . . . . 10 ( 𝐹 ∪ ∅) = 𝐹
94, 7, 83eqtri 2636 . . . . . . . . 9 (𝐹 ∪ {∅}) = 𝐹
103, 9eqtr2i 2633 . . . . . . . 8 𝐹 = 𝐽
11 ufilfil 21518 . . . . . . . . 9 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
12 filunibas 21495 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
1311, 12syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
1410, 13syl5reqr 2659 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋 = 𝐽)
1514sseq2d 3596 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝑋𝑥 𝐽))
161, 15syl5ibr 235 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥𝐽𝑥𝑋))
17 eqid 2610 . . . . . . 7 𝐽 = 𝐽
1817cldss 20643 . . . . . 6 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 𝐽)
1918, 15syl5ibr 235 . . . . 5 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋))
2016, 19jaod 394 . . . 4 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) → 𝑥𝑋))
21 ufilss 21519 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹))
22 ssun1 3738 . . . . . . . . . 10 𝐹 ⊆ (𝐹 ∪ {∅})
2322, 2sseqtr4i 3601 . . . . . . . . 9 𝐹𝐽
2423a1i 11 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐹𝐽)
2524sseld 3567 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥𝐹𝑥𝐽))
2624sseld 3567 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹 → (𝑋𝑥) ∈ 𝐽))
27 filcon 21497 . . . . . . . . . . . . 13 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Con)
28 contop 21030 . . . . . . . . . . . . 13 ((𝐹 ∪ {∅}) ∈ Con → (𝐹 ∪ {∅}) ∈ Top)
2911, 27, 283syl 18 . . . . . . . . . . . 12 (𝐹 ∈ (UFil‘𝑋) → (𝐹 ∪ {∅}) ∈ Top)
302, 29syl5eqel 2692 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → 𝐽 ∈ Top)
3130adantr 480 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
3215biimpa 500 . . . . . . . . . 10 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → 𝑥 𝐽)
3317iscld2 20642 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑥 𝐽) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3431, 32, 33syl2anc 691 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ ( 𝐽𝑥) ∈ 𝐽))
3514difeq1d 3689 . . . . . . . . . . 11 (𝐹 ∈ (UFil‘𝑋) → (𝑋𝑥) = ( 𝐽𝑥))
3635eleq1d 2672 . . . . . . . . . 10 (𝐹 ∈ (UFil‘𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3736adantr 480 . . . . . . . . 9 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐽 ↔ ( 𝐽𝑥) ∈ 𝐽))
3834, 37bitr4d 270 . . . . . . . 8 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → (𝑥 ∈ (Clsd‘𝐽) ↔ (𝑋𝑥) ∈ 𝐽))
3926, 38sylibrd 248 . . . . . . 7 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑥𝑋) → ((𝑋𝑥) ∈ 𝐹𝑥 ∈ (Clsd‘𝐽)))
4025, 39orim12d 879 . . . . . 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 201 . . 3 (𝐹 ∈ (UFil‘𝑋) → ((𝑥𝐽𝑥 ∈ (Clsd‘𝐽)) ↔ 𝑥𝑋))
44 elun 3715 . . 3 (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ (𝑥𝐽𝑥 ∈ (Clsd‘𝐽)))
45 selpw 4115 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
4643, 44, 453bitr4g 302 . 2 (𝐹 ∈ (UFil‘𝑋) → (𝑥 ∈ (𝐽 ∪ (Clsd‘𝐽)) ↔ 𝑥 ∈ 𝒫 𝑋))
4746eqrdv 2608 1 (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024  Filcfil 21459  UFilcufil 21513 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-fbas 19564  df-top 20521  df-cld 20633  df-con 21025  df-fil 21460  df-ufil 21515 This theorem is referenced by: (None)
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