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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfileln0 21701 An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴 ≠ ∅)

Theoremfilsspw 21702 A filter is a subset of the power set of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)

Theoremfilelss 21703 An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → 𝐴𝑋)

Theoremfilss 21704 A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵𝐹)

Theoremfilin 21705 A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ∈ 𝐹)

Theoremfiltop 21706 The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)

Theoremisfil2 21707* Derive the standard axioms of a filter. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) ↔ ((𝐹 ⊆ 𝒫 𝑋 ∧ ¬ ∅ ∈ 𝐹𝑋𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦𝐹 𝑦𝑥𝑥𝐹) ∧ ∀𝑥𝐹𝑦𝐹 (𝑥𝑦) ∈ 𝐹))

Theoremisfildlem 21708* Lemma for isfild 21709. (Contributed by Mario Carneiro, 1-Dec-2013.)
(𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))

Theoremisfild 21709* Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))    &   (𝜑𝐴 ∈ V)    &   (𝜑[𝐴 / 𝑥]𝜓)    &   (𝜑 → ¬ [∅ / 𝑥]𝜓)    &   ((𝜑𝑦𝐴𝑧𝑦) → ([𝑧 / 𝑥]𝜓[𝑦 / 𝑥]𝜓))    &   ((𝜑𝑦𝐴𝑧𝐴) → (([𝑦 / 𝑥]𝜓[𝑧 / 𝑥]𝜓) → [(𝑦𝑧) / 𝑥]𝜓))       (𝜑𝐹 ∈ (Fil‘𝐴))

Theoremfilfi 21710 A filter is closed under taking intersections. (Contributed by Mario Carneiro, 27-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → (fi‘𝐹) = 𝐹)

Theoremfilinn0 21711 The intersection of two elements of a filter can't be empty. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵) ≠ ∅)

Theoremfilintn0 21712 A filter has the finite intersection property. Remark below definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 20-Sep-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)

Theoremfiln0 21713 The empty set is not a filter. Remark below def. 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)

Theoreminfil 21714 The intersection of two filters is a filter. Use fiint 8278 to extend this property to the intersection of a finite set of filters. Paragraph 3 of [BourbakiTop1] p. I.36. (Contributed by FL, 17-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋)) → (𝐹𝐺) ∈ (Fil‘𝑋))

Theoremsnfil 21715 A singleton is a filter. Example 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 16-Sep-2007.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝐵𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))

Theoremfbasweak 21716 A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐹 ⊆ 𝒫 𝑌𝑌𝑉) → 𝐹 ∈ (fBas‘𝑌))

Theoremsnfbas 21717 Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐴𝐵𝐴 ≠ ∅ ∧ 𝐵𝑉) → {𝐴} ∈ (fBas‘𝐵))

Theoremfsubbas 21718 A condition for a set to generate a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝑋𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))

Theoremfbasfip 21719 A filter base has the finite intersection property. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → ¬ ∅ ∈ (fi‘𝐹))

Theoremfbunfip 21720* A helpful lemma for showing that certain sets generate filters. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑌)) → (¬ ∅ ∈ (fi‘(𝐹𝐺)) ↔ ∀𝑥𝐹𝑦𝐺 (𝑥𝑦) ≠ ∅))

Theoremfgval 21721* The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐹 ∩ 𝒫 𝑥) ≠ ∅})

Theoremelfg 21722* A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝐴 ∈ (𝑋filGen𝐹) ↔ (𝐴𝑋 ∧ ∃𝑥𝐹 𝑥𝐴)))

Theoremssfg 21723 A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))

Theoremfgss 21724 A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋) ∧ 𝐹𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))

Theoremfgss2 21725* A condition for a filter to be finer than another involving their filter bases. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐹) ⊆ (𝑋filGen𝐺) ↔ ∀𝑥𝐹𝑦𝐺 𝑦𝑥))

Theoremfgfil 21726 A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)

Theoremelfilss 21727* An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝑋) → (𝐴𝐹 ↔ ∃𝑡𝐹 𝑡𝐴))

Theoremfilfinnfr 21728 No filter containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝑆 ∈ Fin) → 𝐹 ≠ ∅)

Theoremfgcl 21729 A generated filter is a filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (fBas‘𝑋) → (𝑋filGen𝐹) ∈ (Fil‘𝑋))

Theoremfgabs 21730 Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))

Theoremneifil 21731 The neighborhoods of a nonempty set is a filter. Example 2 of [BourbakiTop1] p. I.36. (Contributed by FL, 18-Sep-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))

Theoremfilunibas 21732 Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)

Theoremfilunirn 21733 Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ran Fil ↔ 𝐹 ∈ (Fil‘ 𝐹))

Theoremfilconn 21734 A filter gives rise to a connected topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → (𝐹 ∪ {∅}) ∈ Conn)

Theoremfbasrn 21735* Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐶 = ran (𝑥𝐵 ↦ (𝐹𝑥))       ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋𝑌𝑌𝑉) → 𝐶 ∈ (fBas‘𝑌))

Theoremfiluni 21736* The union of a nonempty set of filters with a common base and closed under pairwise union is a filter. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ⊆ (Fil‘𝑋) ∧ 𝐹 ≠ ∅ ∧ ∀𝑓𝐹𝑔𝐹 (𝑓𝑔) ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Theoremtrfil1 21737 Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → 𝐴 = (𝐿t 𝐴))

Theoremtrfil2 21738* Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by FL, 2-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣𝐿 (𝑣𝐴) ≠ ∅))

Theoremtrfil3 21739 Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑌𝐴) ∈ 𝐿))

Theoremtrfilss 21740 If 𝐴 is a member of the filter, then the filter truncated to 𝐴 is a subset of the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝐹t 𝐴) ⊆ 𝐹)

Theoremfgtr 21741 If 𝐴 is a member of the filter, then truncating 𝐹 to 𝐴 and regenerating the behavior outside 𝐴 using filGen recovers the original filter. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴𝐹) → (𝑋filGen(𝐹t 𝐴)) = 𝐹)

Theoremtrfg 21742 The trace operation and the filGen operation are inverses to one another in some sense, with filGen growing the base set and t shrinking it. See fgtr 21741 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴𝑋𝑋𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)

Theoremtrnei 21743 The trace, over a set 𝐴, of the filter of the neighborhoods of a point 𝑃 is a filter iff 𝑃 belongs to the closure of 𝐴. (This is trfil2 21738 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴𝑌𝑃𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))

Theoremcfinfil 21744* Relative complements of the finite parts of an infinite set is a filter. When 𝐴 = ℕ the set of the relative complements is called Frechet's filter and is used to define the concept of limit of a sequence. (Contributed by FL, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝑋𝑉𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))

Theoremcsdfil 21745* The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Theoremsupfil 21746* The supersets of a nonempty set which are also subsets of a given base set form a filter. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
((𝐴𝑉𝐵𝐴𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴𝐵𝑥} ∈ (Fil‘𝐴))

Theoremzfbas 21747 The set of upper sets of integers is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)
ran ℤ ∈ (fBas‘ℤ)

Theoremuzrest 21748 The restriction of the set of upper sets of integers to an upper set of integers is the set of upper sets of integers based at a point above the cutoff. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (ran ℤt 𝑍) = (ℤ𝑍))

Theoremuzfbas 21749 The set of upper sets of integers based at a point in a fixed upper integer set like is a filter base on , which corresponds to convergence of sequences on . (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)       (𝑀 ∈ ℤ → (ℤ𝑍) ∈ (fBas‘𝑍))

12.2.3  Ultrafilters

Syntaxcufil 21750 Extend class notation with the ultrafilters-on-a-set function.
class UFil

Syntaxcufl 21751 Extend class notation with the ultrafilter lemma.
class UFL

Definitiondf-ufil 21752* Define the set of ultrafilters on a set. An ultrafilter is a filter that gives a definite result for every subset. (Contributed by Jeff Hankins, 30-Nov-2009.)
UFil = (𝑔 ∈ V ↦ {𝑓 ∈ (Fil‘𝑔) ∣ ∀𝑥 ∈ 𝒫 𝑔(𝑥𝑓 ∨ (𝑔𝑥) ∈ 𝑓)})

Definitiondf-ufl 21753* Define the class of base sets for which the ultrafilter lemma filssufil 21763 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓𝑔}

Theoremisufil 21754* The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
(𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋(𝑥𝐹 ∨ (𝑋𝑥) ∈ 𝐹)))

Theoremufilfil 21755 An ultrafilter is a filter. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
(𝐹 ∈ (UFil‘𝑋) → 𝐹 ∈ (Fil‘𝑋))

Theoremufilss 21756 For any subset of the base set of an ultrafilter, either the set is in the ultrafilter or the complement is. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐹 ∨ (𝑋𝑆) ∈ 𝐹))

Theoremufilb 21757 The complement is in an ultrafilter iff the set is not. (Contributed by Mario Carneiro, 11-Dec-2013.) (Revised by Mario Carneiro, 29-Jul-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝑋) → (¬ 𝑆𝐹 ↔ (𝑋𝑆) ∈ 𝐹))

Theoremufilmax 21758 Any filter finer than an ultrafilter is actually equal to it. (Contributed by Jeff Hankins, 1-Dec-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → 𝐹 = 𝐺)

Theoremisufil2 21759* The maximal property of an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (UFil‘𝑋) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐹𝑓𝐹 = 𝑓)))

Theoremufprim 21760 An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐹𝐵𝐹) ↔ (𝐴𝐵) ∈ 𝐹))

Theoremtrufil 21761 Conditions for the trace of an ultrafilter 𝐿 to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
((𝐿 ∈ (UFil‘𝑌) ∧ 𝐴𝑌) → ((𝐿t 𝐴) ∈ (UFil‘𝐴) ↔ 𝐴𝐿))

Theoremfilssufilg 21762* A filter is contained in some ultrafilter. This version of filssufil 21763 contains the choice as a hypothesis (in the assumption that 𝒫 𝒫 𝑋 is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)

Theoremfilssufil 21763* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 9330.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)

Theoremisufl 21764* Define the (strong) ultrafilter lemma, parameterized over base sets. A set 𝑋 satisfies the ultrafilter lemma if every filter on 𝑋 is a subset of some ultrafilter. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑋𝑉 → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓𝑔))

Theoremufli 21765* Property of a set that satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)

Theoremnumufl 21766 Consequence of filssufilg 21762: a set whose double powerset is well-orderable satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝒫 𝒫 𝑋 ∈ dom card → 𝑋 ∈ UFL)

Theoremfiufl 21767 A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑋 ∈ Fin → 𝑋 ∈ UFL)

Theoremacufl 21768 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
(CHOICE → UFL = V)

Theoremssufl 21769 If 𝑌 is a subset of 𝑋 and filters extend to ultrafilters in 𝑋, then they still do in 𝑌. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝑋 ∈ UFL ∧ 𝑌𝑋) → 𝑌 ∈ UFL)

Theoremufileu 21770* If the ultrafilter containing a given filter is unique, the filter is an ultrafilter. (Contributed by Jeff Hankins, 3-Dec-2009.) (Revised by Mario Carneiro, 2-Oct-2015.)
(𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (UFil‘𝑋) ↔ ∃!𝑓 ∈ (UFil‘𝑋)𝐹𝑓))

Theoremfilufint 21771* A filter is equal to the intersection of the ultrafilters containing it. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → {𝑓 ∈ (UFil‘𝑋) ∣ 𝐹𝑓} = 𝐹)

Theoremuffix 21772* Lemma for fixufil 21773 and uffixfr 21774. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝑋𝑉𝐴𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋𝐴𝑥} = (𝑋filGen{{𝐴}})))

Theoremfixufil 21773* 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.)
((𝑋𝑉𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋𝐴𝑥} ∈ (UFil‘𝑋))

Theoremuffixfr 21774* 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.)
(𝐹 ∈ (UFil‘𝑋) → (𝐴 𝐹𝐹 = {𝑥 ∈ 𝒫 𝑋𝐴𝑥}))

Theoremuffix2 21775* A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (UFil‘𝑋) → ( 𝐹 ≠ ∅ ↔ ∃𝑥𝑋 𝐹 = {𝑦 ∈ 𝒫 𝑋𝑥𝑦}))

Theoremuffixsn 21776 The singleton of the generator of a fixed ultrafilter is in the filter. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 𝐹) → {𝐴} ∈ 𝐹)

Theoremufildom1 21777 An ultrafilter is generated by at most one element (because free ultrafilters have no generators and fixed ultrafilters have exactly one). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (UFil‘𝑋) → 𝐹 ≼ 1𝑜)

Theoremuffinfix 21778* An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆𝐹𝑆 ∈ Fin) → ∃𝑥𝑋 𝐹 = {𝑦 ∈ 𝒫 𝑋𝑥𝑦})

Theoremcfinufil 21779* An ultrafilter is free iff it contains the Fréchet filter cfinfil 21744 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
(𝐹 ∈ (UFil‘𝑋) → ( 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ∈ Fin} ⊆ 𝐹))

Theoremufinffr 21780* An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)
((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴𝑓 𝑓 = ∅))

Theoremufilen 21781* Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
(ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋)

Theoremufildr 21782 An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
𝐽 = (𝐹 ∪ {∅})       (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋)

Theoremfin1aufil 21783 There are no definable free ultrafilters in ZFC. However, there are free ultrafilters in some choice-denying constructions. Here we show that given an amorphous set (a.k.a. a Ia-finite I-infinite set) 𝑋, the set of infinite subsets of 𝑋 is a free ultrafilter on 𝑋. (Contributed by Mario Carneiro, 20-May-2015.)
𝐹 = (𝒫 𝑋 ∖ Fin)       (𝑋 ∈ (FinIa ∖ Fin) → (𝐹 ∈ (UFil‘𝑋) ∧ 𝐹 = ∅))

12.2.4  Filter limits

Syntaxcfm 21784 Extend class definition to include the neighborhood filter mapping function.
class FilMap

Syntaxcflim 21785 Extend class notation with a function returning the limit of a filter.
class fLim

Syntaxcflf 21786 Extend class definition to include the function for filter-based function limits.
class fLimf

Syntaxcfcls 21787 Extend class definition to include the cluster point function on filters.
class fClus

Syntaxcfcf 21788 Extend class definition to include the function for cluster points of a function.
class fClusf

Definitiondf-fm 21789* Define a function that takes a filter to a neighborhood filter of the range. (Since we now allow filter bases to have support smaller than the base set, the function has to come first to ensure that curryings are sets.) (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 20-Jul-2015.)
FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡𝑦 ↦ (𝑓𝑡)))))

Definitiondf-flim 21790* Define a function (indexed by a topology 𝑗) whose value is the limits of a filter 𝑓. (Contributed by Jeff Hankins, 4-Sep-2009.)
fLim = (𝑗 ∈ Top, 𝑓 ran Fil ↦ {𝑥 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓𝑓 ⊆ 𝒫 𝑗)})

Definitiondf-flf 21791* Define a function that gives the limits of a function 𝑓 in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.)
fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥𝑚 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))

Definitiondf-fcls 21792* Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.)
fClus = (𝑗 ∈ Top, 𝑓 ran Fil ↦ if( 𝑗 = 𝑓, 𝑥𝑓 ((cls‘𝑗)‘𝑥), ∅))

Definitiondf-fcf 21793* Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.)
fClusf = (𝑗 ∈ Top, 𝑓 ran Fil ↦ (𝑔 ∈ ( 𝑗𝑚 𝑓) ↦ (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓))))

Theoremfmval 21794* Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual positive integer ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦𝐵 ↦ (𝐹𝑦))))

Theoremfmfil 21795 A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋))

Theoremfmf 21796 Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.)
((𝑋𝐴𝑌𝐵𝐹:𝑌𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋))

Theoremfmss 21797 A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
(((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌𝑋𝐵𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶))

Theoremelfm 21798* An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑥𝐵 (𝐹𝑥) ⊆ 𝐴)))

Theoremelfm2 21799* An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐿 = (𝑌filGen𝐵)       ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴𝑋 ∧ ∃𝑥𝐿 (𝐹𝑥) ⊆ 𝐴)))

Theoremfmfg 21800 The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐿 = (𝑌filGen𝐵)       ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))

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