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Theorem fixufil 21773
Description: The condition describing a fixed ultrafilter always produces an ultrafilter. (Contributed by Jeff Hankins, 9-Dec-2009.) (Revised by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 29-Jul-2015.)
Assertion
Ref Expression
fixufil ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋   𝑥,𝑉

Proof of Theorem fixufil
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 uffix 21772 . . . 4 ((𝑋𝑉𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
21simprd 478 . . 3 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
31simpld 474 . . . 4 ((𝑋𝑉𝐴𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4 fgcl 21729 . . . 4 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
53, 4syl 17 . . 3 ((𝑋𝑉𝐴𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
62, 5eqeltrd 2730 . 2 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
7 undif2 4077 . . . . . . . . . 10 (𝑦 ∪ (𝑋𝑦)) = (𝑦𝑋)
8 elpwi 4201 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
9 ssequn1 3816 . . . . . . . . . . 11 (𝑦𝑋 ↔ (𝑦𝑋) = 𝑋)
108, 9sylib 208 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋 → (𝑦𝑋) = 𝑋)
117, 10syl5req 2698 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑋 = (𝑦 ∪ (𝑋𝑦)))
1211eleq2d 2716 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑋𝐴 ∈ (𝑦 ∪ (𝑋𝑦))))
1312biimpac 502 . . . . . . 7 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋𝑦)))
14 elun 3786 . . . . . . 7 (𝐴 ∈ (𝑦 ∪ (𝑋𝑦)) ↔ (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1513, 14sylib 208 . . . . . 6 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1615adantll 750 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
17 ibar 524 . . . . . . 7 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
1817adantl 481 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
19 difss 3770 . . . . . . . . 9 (𝑋𝑦) ⊆ 𝑋
20 elpw2g 4857 . . . . . . . . 9 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
2119, 20mpbiri 248 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
2221ad2antrr 762 . . . . . . 7 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋𝑦) ∈ 𝒫 𝑋)
2322biantrurd 528 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋𝑦) ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2418, 23orbi12d 746 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦𝐴 ∈ (𝑋𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))))
2516, 24mpbid 222 . . . 4 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2625ralrimiva 2995 . . 3 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
27 eleq2 2719 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2827elrab 3396 . . . . 5 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦))
29 eleq2 2719 . . . . . 6 (𝑥 = (𝑋𝑦) → (𝐴𝑥𝐴 ∈ (𝑋𝑦)))
3029elrab 3396 . . . . 5 ((𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))
3128, 30orbi12i 542 . . . 4 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3231ralbii 3009 . . 3 (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3326, 32sylibr 224 . 2 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
34 isufil 21754 . 2 ({𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋) ↔ ({𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})))
356, 33, 34sylanbrc 699 1 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  cun 3605  wss 3607  𝒫 cpw 4191  {csn 4210  cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  Filcfil 21696  UFilcufil 21750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-fg 19792  df-fil 21697  df-ufil 21752
This theorem is referenced by: (None)
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