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Theorem fixufil 21720
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 21719 . . . 4 ((𝑋𝑉𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
21simprd 479 . . 3 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
31simpld 475 . . . 4 ((𝑋𝑉𝐴𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4 fgcl 21676 . . . 4 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
53, 4syl 17 . . 3 ((𝑋𝑉𝐴𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
62, 5eqeltrd 2700 . 2 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
7 undif2 4042 . . . . . . . . . 10 (𝑦 ∪ (𝑋𝑦)) = (𝑦𝑋)
8 elpwi 4166 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
9 ssequn1 3781 . . . . . . . . . . 11 (𝑦𝑋 ↔ (𝑦𝑋) = 𝑋)
108, 9sylib 208 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋 → (𝑦𝑋) = 𝑋)
117, 10syl5req 2668 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑋 = (𝑦 ∪ (𝑋𝑦)))
1211eleq2d 2686 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑋𝐴 ∈ (𝑦 ∪ (𝑋𝑦))))
1312biimpac 503 . . . . . . 7 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋𝑦)))
14 elun 3751 . . . . . . 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 525 . . . . . . 7 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
1817adantl 482 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
19 difss 3735 . . . . . . . . 9 (𝑋𝑦) ⊆ 𝑋
20 elpw2g 4825 . . . . . . . . 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 529 . . . . . 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 2965 . . 3 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
27 eleq2 2689 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2827elrab 3361 . . . . 5 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦))
29 eleq2 2689 . . . . . 6 (𝑥 = (𝑋𝑦) → (𝐴𝑥𝐴 ∈ (𝑋𝑦)))
3029elrab 3361 . . . . 5 ((𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))
3128, 30orbi12i 543 . . . 4 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3231ralbii 2979 . . 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 21701 . 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 698 1 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  cdif 3569  cun 3570  wss 3572  𝒫 cpw 4156  {csn 4175  cfv 5886  (class class class)co 6647  fBascfbas 19728  filGencfg 19729  Filcfil 21643  UFilcufil 21697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-fbas 19737  df-fg 19738  df-fil 21644  df-ufil 21699
This theorem is referenced by: (None)
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