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Theorem fixufil 22005
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 22004 . . . 4 ((𝑋𝑉𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
21simprd 489 . . 3 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
31simpld 488 . . . 4 ((𝑋𝑉𝐴𝑋) → {{𝐴}} ∈ (fBas‘𝑋))
4 fgcl 21961 . . . 4 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
53, 4syl 17 . . 3 ((𝑋𝑉𝐴𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
62, 5eqeltrd 2844 . 2 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
7 undif2 4204 . . . . . . . . . 10 (𝑦 ∪ (𝑋𝑦)) = (𝑦𝑋)
8 elpwi 4325 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
9 ssequn1 3945 . . . . . . . . . . 11 (𝑦𝑋 ↔ (𝑦𝑋) = 𝑋)
108, 9sylib 209 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋 → (𝑦𝑋) = 𝑋)
117, 10syl5req 2812 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝑋𝑋 = (𝑦 ∪ (𝑋𝑦)))
1211eleq2d 2830 . . . . . . . 8 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑋𝐴 ∈ (𝑦 ∪ (𝑋𝑦))))
1312biimpac 470 . . . . . . 7 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → 𝐴 ∈ (𝑦 ∪ (𝑋𝑦)))
14 elun 3915 . . . . . . 7 (𝐴 ∈ (𝑦 ∪ (𝑋𝑦)) ↔ (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1513, 14sylib 209 . . . . . 6 ((𝐴𝑋𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
1615adantll 705 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦𝐴 ∈ (𝑋𝑦)))
17 ibar 524 . . . . . . 7 (𝑦 ∈ 𝒫 𝑋 → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
1817adantl 473 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴𝑦 ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦)))
19 difss 3899 . . . . . . . . 9 (𝑋𝑦) ⊆ 𝑋
20 elpw2g 4985 . . . . . . . . 9 (𝑋𝑉 → ((𝑋𝑦) ∈ 𝒫 𝑋 ↔ (𝑋𝑦) ⊆ 𝑋))
2119, 20mpbiri 249 . . . . . . . 8 (𝑋𝑉 → (𝑋𝑦) ∈ 𝒫 𝑋)
2221ad2antrr 717 . . . . . . 7 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝑋𝑦) ∈ 𝒫 𝑋)
2322biantrurd 528 . . . . . 6 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → (𝐴 ∈ (𝑋𝑦) ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2418, 23orbi12d 942 . . . . 5 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦𝐴 ∈ (𝑋𝑦)) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))))
2516, 24mpbid 223 . . . 4 (((𝑋𝑉𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
2625ralrimiva 3113 . . 3 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
27 eleq2 2833 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
2827elrab 3519 . . . . 5 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑦 ∈ 𝒫 𝑋𝐴𝑦))
29 eleq2 2833 . . . . . 6 (𝑥 = (𝑋𝑦) → (𝐴𝑥𝐴 ∈ (𝑋𝑦)))
3029elrab 3519 . . . . 5 ((𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦)))
3128, 30orbi12i 938 . . . 4 ((𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3231ralbii 3127 . . 3 (∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) ↔ ∀𝑦 ∈ 𝒫 𝑋((𝑦 ∈ 𝒫 𝑋𝐴𝑦) ∨ ((𝑋𝑦) ∈ 𝒫 𝑋𝐴 ∈ (𝑋𝑦))))
3326, 32sylibr 225 . 2 ((𝑋𝑉𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋(𝑦 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∨ (𝑋𝑦) ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
34 isufil 21986 . 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 578 1 ((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wral 3055  {crab 3059  cdif 3729  cun 3730  wss 3732  𝒫 cpw 4315  {csn 4334  cfv 6068  (class class class)co 6842  fBascfbas 20007  filGencfg 20008  Filcfil 21928  UFilcufil 21982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-fbas 20016  df-fg 20017  df-fil 21929  df-ufil 21984
This theorem is referenced by: (None)
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