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Theorem uffixfr 21928
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 474 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2 ufilfil 21909 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filtop 21860 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
42, 3syl 17 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝑋𝐹)
54adantr 472 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝑋𝐹)
6 filn0 21867 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
7 intssuni 4651 . . . . . . . . 9 (𝐹 ≠ ∅ → 𝐹 𝐹)
82, 6, 73syl 18 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
9 filunibas 21886 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
102, 9syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
118, 10sseqtrd 3782 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
1211sselda 3744 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
13 uffix 21926 . . . . . 6 ((𝑋𝐹𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
145, 12, 13syl2anc 696 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
1514simprd 482 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
1614simpld 477 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
17 fgcl 21883 . . . . 5 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1816, 17syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1915, 18eqeltrd 2839 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
202adantr 472 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (Fil‘𝑋))
21 filsspw 21856 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2220, 21syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
23 elintg 4635 . . . . . 6 (𝐴 𝐹 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
2423ibi 256 . . . . 5 (𝐴 𝐹 → ∀𝑥𝐹 𝐴𝑥)
2524adantl 473 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ∀𝑥𝐹 𝐴𝑥)
26 ssrab 3821 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝐹 ⊆ 𝒫 𝑋 ∧ ∀𝑥𝐹 𝐴𝑥))
2722, 25, 26sylanbrc 701 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
28 ufilmax 21912 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
291, 19, 27, 28syl3anc 1477 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
30 eqimss 3798 . . . . 5 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3130adantl 473 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3226simprbi 483 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → ∀𝑥𝐹 𝐴𝑥)
3331, 32syl 17 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → ∀𝑥𝐹 𝐴𝑥)
34 eleq2 2828 . . . . . 6 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → (𝑋𝐹𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
3534biimpac 504 . . . . 5 ((𝑋𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
364, 35sylan 489 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
37 eleq2 2828 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥𝐴𝑋))
3837elrab 3504 . . . . 5 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑋 ∈ 𝒫 𝑋𝐴𝑋))
3938simprbi 483 . . . 4 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐴𝑋)
40 elintg 4635 . . . 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 913 1 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  {crab 3054  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588   cint 4627  cfv 6049  (class class class)co 6813  fBascfbas 19936  filGencfg 19937  Filcfil 21850  UFilcufil 21904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-fbas 19945  df-fg 19946  df-fil 21851  df-ufil 21906
This theorem is referenced by:  uffix2  21929  uffixsn  21930
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