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Theorem fixufil 21536
 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 21535 . . . 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 21492 . . . 4 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
53, 4syl 17 . . 3 ((𝑋𝑉𝐴𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
62, 5eqeltrd 2688 . 2 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
7 undif2 3996 . . . . . . . . . 10 (𝑦 ∪ (𝑋𝑦)) = (𝑦𝑋)
8 elpwi 4117 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
9 ssequn1 3745 . . . . . . . . . . 11 (𝑦𝑋 ↔ (𝑦𝑋) = 𝑋)
108, 9sylib 207 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋 → (𝑦𝑋) = 𝑋)
117, 10syl5req 2657 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑋 = (𝑦 ∪ (𝑋𝑦)))
1211eleq2d 2673 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑋𝐴 ∈ (𝑦 ∪ (𝑋𝑦))))
1312biimpac 502 . . . . . . 7 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋𝑦)))
14 elun 3715 . . . . . . 7 (𝐴 ∈ (𝑦 ∪ (𝑋𝑦)) ↔ (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1513, 14sylib 207 . . . . . 6 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1615adantll 746 . . . . 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 3699 . . . . . . . . 9 (𝑋𝑦) ⊆ 𝑋
20 elpw2g 4754 . . . . . . . . 9 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
2119, 20mpbiri 247 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
2221ad2antrr 758 . . . . . . 7 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋𝑦) ∈ 𝒫 𝑋)
2322biantrurd 528 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋𝑦) ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2418, 23orbi12d 742 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦𝐴 ∈ (𝑋𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))))
2516, 24mpbid 221 . . . 4 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2625ralrimiva 2949 . . 3 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
27 eleq2 2677 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2827elrab 3331 . . . . 5 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦))
29 eleq2 2677 . . . . . 6 (𝑥 = (𝑋𝑦) → (𝐴𝑥𝐴 ∈ (𝑋𝑦)))
3029elrab 3331 . . . . 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 2963 . . 3 (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3326, 32sylibr 223 . 2 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
34 isufil 21517 . 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 695 1 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ‘cfv 5804  (class class class)co 6549  fBascfbas 19555  filGencfg 19556  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 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-int 4411  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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fbas 19564  df-fg 19565  df-fil 21460  df-ufil 21515 This theorem is referenced by: (None)
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