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Theorem fixufil 23426
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 23425 . . . 4 ((𝑋𝑉𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
21simprd 497 . . 3 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
31simpld 496 . . . 4 ((𝑋𝑉𝐴𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4 fgcl 23382 . . . 4 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
53, 4syl 17 . . 3 ((𝑋𝑉𝐴𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
62, 5eqeltrd 2834 . 2 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
7 undif2 4477 . . . . . . . . . 10 (𝑦 ∪ (𝑋𝑦)) = (𝑦𝑋)
8 elpwi 4610 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
9 ssequn1 4181 . . . . . . . . . . 11 (𝑦𝑋 ↔ (𝑦𝑋) = 𝑋)
108, 9sylib 217 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋 → (𝑦𝑋) = 𝑋)
117, 10eqtr2id 2786 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑋 = (𝑦 ∪ (𝑋𝑦)))
1211eleq2d 2820 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑋𝐴 ∈ (𝑦 ∪ (𝑋𝑦))))
1312biimpac 480 . . . . . . 7 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋𝑦)))
14 elun 4149 . . . . . . 7 (𝐴 ∈ (𝑦 ∪ (𝑋𝑦)) ↔ (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1513, 14sylib 217 . . . . . 6 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1615adantll 713 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
17 ibar 530 . . . . . . 7 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
1817adantl 483 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
19 difss 4132 . . . . . . . . 9 (𝑋𝑦) ⊆ 𝑋
20 elpw2g 5345 . . . . . . . . 9 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
2119, 20mpbiri 258 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
2221ad2antrr 725 . . . . . . 7 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋𝑦) ∈ 𝒫 𝑋)
2322biantrurd 534 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋𝑦) ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2418, 23orbi12d 918 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦𝐴 ∈ (𝑋𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))))
2516, 24mpbid 231 . . . 4 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2625ralrimiva 3147 . . 3 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
27 eleq2 2823 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2827elrab 3684 . . . . 5 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦))
29 eleq2 2823 . . . . . 6 (𝑥 = (𝑋𝑦) → (𝐴𝑥𝐴 ∈ (𝑋𝑦)))
3029elrab 3684 . . . . 5 ((𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))
3128, 30orbi12i 914 . . . 4 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3231ralbii 3094 . . 3 (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3326, 32sylibr 233 . 2 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
34 isufil 23407 . 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 584 1 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  {crab 3433  cdif 3946  cun 3947  wss 3949  𝒫 cpw 4603  {csn 4629  cfv 6544  (class class class)co 7409  fBascfbas 20932  filGencfg 20933  Filcfil 23349  UFilcufil 23403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-fg 20942  df-fil 23350  df-ufil 23405
This theorem is referenced by: (None)
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