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Theorem uffixfr 22219
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 483 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (UFil‘𝑋))
2 ufilfil 22200 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))
3 filtop 22151 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
42, 3syl 17 . . . . . 6 (𝐹 ∈ (UFil‘𝑋) → 𝑋𝐹)
5 filn0 22158 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
6 intssuni 4810 . . . . . . . . 9 (𝐹 ≠ ∅ → 𝐹 𝐹)
72, 5, 63syl 18 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 𝐹)
8 filunibas 22177 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
92, 8syl 17 . . . . . . . 8 (𝐹 ∈ (UFil‘𝑋) → 𝐹 = 𝑋)
107, 9sseqtrd 3934 . . . . . . 7 (𝐹 ∈ (UFil‘𝑋) → 𝐹𝑋)
1110sselda 3895 . . . . . 6 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐴𝑋)
12 uffix 22217 . . . . . 6 ((𝑋𝐹𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
134, 11, 12syl2an2r 681 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))
1413simprd 496 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}}))
1513simpld 495 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {{𝐴}} ∈ (fBas‘𝑋))
16 fgcl 22174 . . . . 5 ({{𝐴}} ∈ (fBas‘𝑋) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1715, 16syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → (𝑋filGen{{𝐴}}) ∈ (Fil‘𝑋))
1814, 17eqeltrd 2885 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋))
192adantr 481 . . . . 5 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ∈ (Fil‘𝑋))
20 filsspw 22147 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2119, 20syl 17 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 ⊆ 𝒫 𝑋)
22 elintg 4796 . . . . . 6 (𝐴 𝐹 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
2322ibi 268 . . . . 5 (𝐴 𝐹 → ∀𝑥𝐹 𝐴𝑥)
2423adantl 482 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → ∀𝑥𝐹 𝐴𝑥)
25 ssrab 3976 . . . 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 22203 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
281, 18, 26, 27syl3anc 1364 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
29 eqimss 3950 . . . . 5 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3029adantl 482 . . . 4 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥})
3125simprbi 497 . . . 4 (𝐹 ⊆ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → ∀𝑥𝐹 𝐴𝑥)
3230, 31syl 17 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → ∀𝑥𝐹 𝐴𝑥)
33 eleq2 2873 . . . . . 6 (𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → (𝑋𝐹𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
3433biimpac 479 . . . . 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 2873 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝑥𝐴𝑋))
3736elrab 3621 . . . . 5 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ↔ (𝑋 ∈ 𝒫 𝑋𝐴𝑋))
3837simprbi 497 . . . 4 (𝑋 ∈ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} → 𝐴𝑋)
39 elintg 4796 . . . 4 (𝐴𝑋 → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4035, 38, 393syl 18 . . 3 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → (𝐴 𝐹 ↔ ∀𝑥𝐹 𝐴𝑥))
4132, 40mpbird 258 . 2 ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}) → 𝐴 𝐹)
4228, 41impbida 797 1 (𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wne 2986  wral 3107  {crab 3111  wss 3865  c0 4217  𝒫 cpw 4459  {csn 4478   cuni 4751   cint 4788  cfv 6232  (class class class)co 7023  fBascfbas 20219  filGencfg 20220  Filcfil 22141  UFilcufil 22195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-int 4789  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-fbas 20228  df-fg 20229  df-fil 22142  df-ufil 22197
This theorem is referenced by:  uffix2  22220  uffixsn  22221
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