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Theorem uffixfr 21774
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 472 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2 ufilfil 21755 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filtop 21706 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
42, 3syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋𝐹)
54adantr 480 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝑋𝐹)
6 filn0 21713 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
7 intssuni 4531 . . . . . . . . 9 (𝐹 ≠ ∅ → 𝐹 𝐹)
82, 6, 73syl 18 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
9 filunibas 21732 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
102, 9syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
118, 10sseqtrd 3674 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
1211sselda 3636 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
13 uffix 21772 . . . . . 6 ((𝑋𝐹𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
145, 12, 13syl2anc 694 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
1514simprd 478 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
1614simpld 474 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
17 fgcl 21729 . . . . 5 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1816, 17syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1915, 18eqeltrd 2730 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
202adantr 480 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (Fil‘𝑋))
21 filsspw 21702 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2220, 21syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
23 elintg 4515 . . . . . 6 (𝐴 𝐹 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
2423ibi 256 . . . . 5 (𝐴 𝐹 → ∀𝑥𝐹 𝐴𝑥)
2524adantl 481 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ∀𝑥𝐹 𝐴𝑥)
26 ssrab 3713 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥𝐹 𝐴𝑥))
2722, 25, 26sylanbrc 699 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
28 ufilmax 21758 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
291, 19, 27, 28syl3anc 1366 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
30 eqimss 3690 . . . . 5 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3130adantl 481 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3226simprbi 479 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → ∀𝑥𝐹 𝐴𝑥)
3331, 32syl 17 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → ∀𝑥𝐹 𝐴𝑥)
34 eleq2 2719 . . . . . 6 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → (𝑋𝐹𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
3534biimpac 502 . . . . 5 ((𝑋𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
364, 35sylan 487 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
37 eleq2 2719 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥𝐴𝑋))
3837elrab 3396 . . . . 5 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑋 ∈ 𝒫 𝑋𝐴𝑋))
3938simprbi 479 . . . 4 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐴𝑋)
40 elintg 4515 . . . 4 (𝐴𝑋 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4136, 39, 403syl 18 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4233, 41mpbird 247 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐴 𝐹)
4329, 42impbida 895 1 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  {crab 2945  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468   cint 4507  cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  Filcfil 21696  UFilcufil 21750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-fg 19792  df-fil 21697  df-ufil 21752
This theorem is referenced by:  uffix2  21775  uffixsn  21776
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