P. 231 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fbfinnfr 23001	No filter base containing a finite element is free. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∩ 𝐹 ≠ ∅)
Theorem	opnfbas 23002*	The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋))
Theorem	trfbas2 23003	Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ¬ ∅ ∈ (𝐹 ↾t 𝐴)))
Theorem	trfbas 23004*	Conditions for the trace of a filter base 𝐹 to be a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐹 ↾t 𝐴) ∈ (fBas‘𝐴) ↔ ∀𝑣 ∈ 𝐹 (𝑣 ∩ 𝐴) ≠ ∅))
12.2.2  Filters
Syntax	cfil 23005	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 23006*	The set of filters on a set. Definition 1 (axioms FI, FIIa, FIIb, FIII) of [BourbakiTop1] p. I.36. Filters are used to define the concept of limit in the general case. They are a generalization of the idea of neighborhoods. Suppose you are in ℝ. With neighborhoods you can express the idea of a variable that tends to a specific number but you can't express the idea of a variable that tends to infinity. Filters relax the "axioms" of neighborhoods and then succeed in expressing the idea of something that tends to infinity. Filters were invented by Cartan in 1937 and made famous by Bourbaki in his treatise. A notion similar to the notion of filter is the concept of net invented by Moore and Smith in 1922. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ Fil = (𝑧 ∈ V ↦ {𝑓 ∈ (fBas‘𝑧) ∣ ∀𝑥 ∈ 𝒫 𝑧((𝑓 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝑓)})
Theorem	isfil 23007*	The predicate "is a filter." (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥 ∈ 𝐹)))
Theorem	filfbas 23008	A filter is a filter base. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
Theorem	0nelfil 23009	The empty set doesn't belong to a filter. (Contributed by FL, 20-Jul-2007.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∅ ∈ 𝐹)
Theorem	fileln0 23010	An element of a filter is nonempty. (Contributed by FL, 24-May-2011.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ≠ ∅)
Theorem	filsspw 23011	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)
Theorem	filelss 23012	An element of a filter is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹) → 𝐴 ⊆ 𝑋)
Theorem	filss 23013	A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ⊆ 𝑋 ∧ 𝐴 ⊆ 𝐵)) → 𝐵 ∈ 𝐹)
Theorem	filin 23014	A filter is closed under taking intersections. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ∈ 𝐹)
Theorem	filtop 23015	The underlying set belongs to the filter. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
Theorem	isfil2 23016*	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 ∧ ¬ ∅ ∈ 𝐹 ∧ 𝑋 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝒫 𝑋(∃𝑦 ∈ 𝐹 𝑦 ⊆ 𝑥 → 𝑥 ∈ 𝐹) ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝑥 ∩ 𝑦) ∈ 𝐹))
Theorem	isfildlem 23017*	Lemma for isfild 23018. (Contributed by Mario Carneiro, 1-Dec-2013.)
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐵 ∈ 𝐹 ↔ (𝐵 ⊆ 𝐴 ∧ [𝐵 / 𝑥]𝜓)))
Theorem	isfild 23018*	Sufficient condition for a set of the form {𝑥 ∈ 𝒫 𝐴 ∣ 𝜑} to be a filter. (Contributed by Mario Carneiro, 1-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.) (Revised by AV, 10-Apr-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐹 ↔ (𝑥 ⊆ 𝐴 ∧ 𝜓)))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → [𝐴 / 𝑥]𝜓)    &   ⊢ (𝜑 → ¬ [∅ / 𝑥]𝜓)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝑦) → ([𝑧 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜓))    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴) → (([𝑦 / 𝑥]𝜓 ∧ [𝑧 / 𝑥]𝜓) → [(𝑦 ∩ 𝑧) / 𝑥]𝜓))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (Fil‘𝐴))
Theorem	filfi 23019	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)
Theorem	filinn0 23020	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) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ∩ 𝐵) ≠ ∅)
Theorem	filintn0 23021	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) ∧ (𝐴 ⊆ 𝐹 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin)) → ∩ 𝐴 ≠ ∅)
Theorem	filn0 23022	The empty set is not a filter. Remark below Definition 1 of [BourbakiTop1] p. I.36. (Contributed by FL, 30-Oct-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
Theorem	infil 23023	The intersection of two filters is a filter. Use fiint 9100 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)) → (𝐹 ∩ 𝐺) ∈ (Fil‘𝑋))
Theorem	snfil 23024	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.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅) → {𝐴} ∈ (Fil‘𝐴))
Theorem	fbasweak 23025	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 𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐹 ∈ (fBas‘𝑌))
Theorem	snfbas 23026	Condition for a singleton to be a filter base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ 𝑉) → {𝐴} ∈ (fBas‘𝐵))
Theorem	fsubbas 23027	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.)
⊢ (𝑋 ∈ 𝑉 → ((fi‘𝐴) ∈ (fBas‘𝑋) ↔ (𝐴 ⊆ 𝒫 𝑋 ∧ 𝐴 ≠ ∅ ∧ ¬ ∅ ∈ (fi‘𝐴))))
Theorem	fbasfip 23028	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))
Theorem	fbunfip 23029*	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‘(𝐹 ∪ 𝐺)) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 (𝑥 ∩ 𝑦) ≠ ∅))
Theorem	fgval 23030*	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) ≠ ∅})
Theorem	elfg 23031*	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) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)))
Theorem	ssfg 23032	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))
Theorem	fgss 23033	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) ∧ 𝐹 ⊆ 𝐺) → (𝑋filGen𝐹) ⊆ (𝑋filGen𝐺))
Theorem	fgss2 23034*	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) ↔ ∀𝑥 ∈ 𝐹 ∃𝑦 ∈ 𝐺 𝑦 ⊆ 𝑥))
Theorem	fgfil 23035	A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
Theorem	elfilss 23036*	An element belongs to a filter iff any element below it does. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑡 ∈ 𝐹 𝑡 ⊆ 𝐴))
Theorem	filfinnfr 23037	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) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∩ 𝐹 ≠ ∅)
Theorem	fgcl 23038	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))
Theorem	fgabs 23039	Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
Theorem	neifil 23040	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) ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((nei‘𝐽)‘𝑆) ∈ (Fil‘𝑋))
Theorem	filunibas 23041	Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
Theorem	filunirn 23042	Two ways to express a filter on an unspecified base. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ ∪ ran Fil ↔ 𝐹 ∈ (Fil‘∪ 𝐹))
Theorem	filconn 23043	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)
Theorem	fbasrn 23044*	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 (𝑥 ∈ 𝐵 ↦ (𝐹 “ 𝑥))    ⇒   ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹:𝑋⟶𝑌 ∧ 𝑌 ∈ 𝑉) → 𝐶 ∈ (fBas‘𝑌))
Theorem	filuni 23045*	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 ≠ ∅ ∧ ∀𝑓 ∈ 𝐹 ∀𝑔 ∈ 𝐹 (𝑓 ∪ 𝑔) ∈ 𝐹) → ∪ 𝐹 ∈ (Fil‘𝑋))
Theorem	trfil1 23046	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) ∧ 𝐴 ⊆ 𝑌) → 𝐴 = ∪ (𝐿 ↾t 𝐴))
Theorem	trfil2 23047*	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) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ∀𝑣 ∈ 𝐿 (𝑣 ∩ 𝐴) ≠ ∅))
Theorem	trfil3 23048	Conditions for the trace of a filter 𝐿 to be a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐿 ∈ (Fil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (Fil‘𝐴) ↔ ¬ (𝑌 ∖ 𝐴) ∈ 𝐿))
Theorem	trfilss 23049	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) ∧ 𝐴 ∈ 𝐹) → (𝐹 ↾t 𝐴) ⊆ 𝐹)
Theorem	fgtr 23050	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) ∧ 𝐴 ∈ 𝐹) → (𝑋filGen(𝐹 ↾t 𝐴)) = 𝐹)
Theorem	trfg 23051	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 23050 for the converse cancellation law. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐹 ∈ (Fil‘𝐴) ∧ 𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ 𝑉) → ((𝑋filGen𝐹) ↾t 𝐴) = 𝐹)
Theorem	trnei 23052	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 23047 applied to a filter of neighborhoods.) (Contributed by FL, 15-Sep-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑌) ∧ 𝐴 ⊆ 𝑌 ∧ 𝑃 ∈ 𝑌) → (𝑃 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑃}) ↾t 𝐴) ∈ (Fil‘𝐴)))
Theorem	cfinfil 23053*	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.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))
Theorem	csdfil 23054*	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 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Theorem	supfil 23055*	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝐵 ⊆ 𝑥} ∈ (Fil‘𝐴))
Theorem	zfbas 23056	The set of upper sets of integers is a filter base on ℤ, which corresponds to convergence of sequences on ℤ. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ ran ℤ≥ ∈ (fBas‘ℤ)
Theorem	uzrest 23057	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 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → (ran ℤ≥ ↾t 𝑍) = (ℤ≥ “ 𝑍))
Theorem	uzfbas 23058	The set of upper sets of integers based at a point in a fixed upper integer set like ℕ is a filter base on ℕ, which corresponds to convergence of sequences on ℕ. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ 𝑍 = (ℤ≥‘𝑀)    ⇒   ⊢ (𝑀 ∈ ℤ → (ℤ≥ “ 𝑍) ∈ (fBas‘𝑍))
12.2.3  Ultrafilters
Syntax	cufil 23059	Extend class notation with the ultrafilters-on-a-set function.
class UFil
Syntax	cufl 23060	Extend class notation with the ultrafilter lemma.
class UFL
Definition	df-ufil 23061*	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(𝑥 ∈ 𝑓 ∨ (𝑔 ∖ 𝑥) ∈ 𝑓)})
Definition	df-ufl 23062*	Define the class of base sets for which the ultrafilter lemma filssufil 23072 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
Theorem	isufil 23063*	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(𝑥 ∈ 𝐹 ∨ (𝑋 ∖ 𝑥) ∈ 𝐹)))
Theorem	ufilfil 23064	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))
Theorem	ufilss 23065	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) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐹 ∨ (𝑋 ∖ 𝑆) ∈ 𝐹))
Theorem	ufilb 23066	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) ∧ 𝑆 ⊆ 𝑋) → (¬ 𝑆 ∈ 𝐹 ↔ (𝑋 ∖ 𝑆) ∈ 𝐹))
Theorem	ufilmax 23067	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) ∧ 𝐹 ⊆ 𝐺) → 𝐹 = 𝐺)
Theorem	isufil2 23068*	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)(𝐹 ⊆ 𝑓 → 𝐹 = 𝑓)))
Theorem	ufprim 23069	An ultrafilter is a prime filter. (Contributed by Jeff Hankins, 1-Jan-2010.) (Revised by Mario Carneiro, 2-Aug-2015.)
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑋) → ((𝐴 ∈ 𝐹 ∨ 𝐵 ∈ 𝐹) ↔ (𝐴 ∪ 𝐵) ∈ 𝐹))
Theorem	trufil 23070	Conditions for the trace of an ultrafilter 𝐿 to be an ultrafilter. (Contributed by Mario Carneiro, 27-Aug-2015.)
⊢ ((𝐿 ∈ (UFil‘𝑌) ∧ 𝐴 ⊆ 𝑌) → ((𝐿 ↾t 𝐴) ∈ (UFil‘𝐴) ↔ 𝐴 ∈ 𝐿))
Theorem	filssufilg 23071*	A filter is contained in some ultrafilter. This version of filssufil 23072 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)𝐹 ⊆ 𝑓)
Theorem	filssufil 23072*	A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 10235.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
Theorem	isufl 23073*	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.)
⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ UFL ↔ ∀𝑓 ∈ (Fil‘𝑋)∃𝑔 ∈ (UFil‘𝑋)𝑓 ⊆ 𝑔))
Theorem	ufli 23074*	Property of a set that satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝑋 ∈ UFL ∧ 𝐹 ∈ (Fil‘𝑋)) → ∃𝑓 ∈ (UFil‘𝑋)𝐹 ⊆ 𝑓)
Theorem	numufl 23075	Consequence of filssufilg 23071: 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)
Theorem	fiufl 23076	A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑋 ∈ Fin → 𝑋 ∈ UFL)
Theorem	acufl 23077	The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (CHOICE → UFL = V)
Theorem	ssufl 23078	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 ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ UFL)
Theorem	ufileu 23079*	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)𝐹 ⊆ 𝑓))
Theorem	filufint 23080*	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) ∣ 𝐹 ⊆ 𝑓} = 𝐹)
Theorem	uffix 23081*	Lemma for fixufil 23082 and uffixfr 23083. (Contributed by Mario Carneiro, 12-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → ({{𝐴}} ∈ (fBas‘𝑋) ∧ {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} = (𝑋filGen{{𝐴}})))
Theorem	fixufil 23082*	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.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥} ∈ (UFil‘𝑋))
Theorem	uffixfr 23083*	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 ∈ ∩ 𝐹 ↔ 𝐹 = {𝑥 ∈ 𝒫 𝑋 ∣ 𝐴 ∈ 𝑥}))
Theorem	uffix2 23084*	A classification of fixed ultrafilters. (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 ≠ ∅ ↔ ∃𝑥 ∈ 𝑋 𝐹 = {𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦}))
Theorem	uffixsn 23085	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)
Theorem	ufildom1 23086	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)
Theorem	uffinfix 23087*	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) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝐹 = {𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦})
Theorem	cfinufil 23088*	An ultrafilter is free iff it contains the Fréchet filter cfinfil 23053 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹))
Theorem	ufinffr 23089*	An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅))
Theorem	ufilen 23090*	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)∀𝑥 ∈ 𝑓 𝑥 ≈ 𝑋)
Theorem	ufildr 23091	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)
Theorem	fin1aufil 23092	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
Syntax	cfm 23093	Extend class definition to include the neighborhood filter mapping function.
class FilMap
Syntax	cflim 23094	Extend class notation with a function returning the limit of a filter.
class fLim
Syntax	cflf 23095	Extend class definition to include the function for filter-based function limits.
class fLimf
Syntax	cfcls 23096	Extend class definition to include the cluster point function on filters.
class fClus
Syntax	cfcf 23097	Extend class definition to include the function for cluster points of a function.
class fClusf
Definition	df-fm 23098*	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 (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡)))))
Definition	df-flim 23099*	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}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)})
Definition	df-flf 23100*	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 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))