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Theorem uffixfr 23931
Description: An ultrafilter is either fixed or free. A fixed ultrafilter is called principal (generated by a single element 𝐴), and a free ultrafilter is called nonprincipal (having empty intersection). Note that examples of free ultrafilters cannot be defined in ZFC without some form of global choice. (Contributed by Jeff Hankins, 4-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
uffixfr (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑋

Proof of Theorem uffixfr
StepHypRef Expression
1 simpl 482 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2 ufilfil 23912 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filtop 23863 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
42, 3syl 17 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝑋𝐹)
5 filn0 23870 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
6 intssuni 4970 . . . . . . . . 9 (𝐹 ≠ ∅ → 𝐹 𝐹)
72, 5, 63syl 18 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
8 filunibas 23889 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
92, 8syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
107, 9sseqtrd 4020 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
1110sselda 3983 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
12 uffix 23929 . . . . . 6 ((𝑋𝐹𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
134, 11, 12syl2an2r 685 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
1413simprd 495 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
1513simpld 494 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
16 fgcl 23886 . . . . 5 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1814, 17eqeltrd 2841 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
192adantr 480 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (Fil‘𝑋))
20 filsspw 23859 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2119, 20syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
22 elintg 4954 . . . . . 6 (𝐴 𝐹 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
2322ibi 267 . . . . 5 (𝐴 𝐹 → ∀𝑥𝐹 𝐴𝑥)
2423adantl 481 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ∀𝑥𝐹 𝐴𝑥)
25 ssrab 4073 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥𝐹 𝐴𝑥))
2621, 24, 25sylanbrc 583 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
27 ufilmax 23915 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
281, 18, 26, 27syl3anc 1373 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
29 eqimss 4042 . . . . 5 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3029adantl 481 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3125simprbi 496 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → ∀𝑥𝐹 𝐴𝑥)
3230, 31syl 17 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → ∀𝑥𝐹 𝐴𝑥)
33 eleq2 2830 . . . . . 6 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → (𝑋𝐹𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
3433biimpac 478 . . . . 5 ((𝑋𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
354, 34sylan 580 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
36 eleq2 2830 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥𝐴𝑋))
3736elrab 3692 . . . . 5 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑋 ∈ 𝒫 𝑋𝐴𝑋))
3837simprbi 496 . . . 4 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐴𝑋)
39 elintg 4954 . . . 4 (𝐴𝑋 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4035, 38, 393syl 18 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4132, 40mpbird 257 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐴 𝐹)
4228, 41impbida 801 1 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  {crab 3436  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907   cint 4946  cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  Filcfil 23853  UFilcufil 23907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fil 23854  df-ufil 23909
This theorem is referenced by:  uffix2  23932  uffixsn  23933
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