P. 238 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xkocnv 23701*	The inverse of the "currying" function 𝐹 is the uncurrying function. (Contributed by Mario Carneiro, 13-Apr-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐿 ∈ Top)    ⇒   ⊢ (𝜑 → ◡𝐹 = (𝑔 ∈ (𝐽 Cn (𝐿 ↑ko 𝐾)) ↦ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ((𝑔‘𝑥)‘𝑦))))
Theorem	xkohmeo 23702*	The Exponential Law for topological spaces. The "currying" function 𝐹 is a homeomorphism on function spaces when 𝐽 and 𝐾 are exponentiable spaces (by xkococn 23547, it is sufficient to assume that 𝐽, 𝐾 are locally compact to ensure exponentiability). (Contributed by Mario Carneiro, 13-Apr-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ 𝐹 = (𝑓 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↦ (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ (𝑥𝑓𝑦))))    &   ⊢ (𝜑 → 𝐽 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Comp)    &   ⊢ (𝜑 → 𝐿 ∈ Top)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐿 ↑ko (𝐽 ×t 𝐾))Homeo((𝐿 ↑ko 𝐾) ↑ko 𝐽)))
Theorem	qtopf1 23703	If a quotient map is injective, then it is a homeomorphism. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–1-1→𝑌)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
Theorem	qtophmeo 23704*	If two functions on a base topology 𝐽 make the same identifications in order to create quotient spaces 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺, then not only are 𝐽 qTop 𝐹 and 𝐽 qTop 𝐺 homeomorphic, but there is a unique homeomorphism that makes the diagram commute. (Contributed by Mario Carneiro, 24-Mar-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)    &   ⊢ (𝜑 → 𝐺:𝑋–onto→𝑌)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐺‘𝑥) = (𝐺‘𝑦)))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ ((𝐽 qTop 𝐹)Homeo(𝐽 qTop 𝐺))𝐺 = (𝑓 ∘ 𝐹))
Theorem	t0kq 23705*	A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})    ⇒   ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))
Theorem	kqhmph 23706	A topological space is T0 iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))
Theorem	ist1-5lem 23707	Lemma for ist1-5 23709 and similar theorems. If 𝐴 is a topological property which implies T0, such as T1 or T2, the property can be "decomposed" into T0 and a non-T0 version of property 𝐴 (which is defined as stating that the Kolmogorov quotient of the space has property 𝐴). For example, if 𝐴 is T1, then the theorem states that a space is T1 iff it is T0 and its Kolmogorov quotient is T1 (we call this property R0). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Kol2)    &   ⊢ (𝐽 ≃ (KQ‘𝐽) → (𝐽 ∈ 𝐴 → (KQ‘𝐽) ∈ 𝐴))    &   ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ 𝐴 → 𝐽 ∈ 𝐴))    ⇒   ⊢ (𝐽 ∈ 𝐴 ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ 𝐴))
Theorem	t1r0 23708	A T1 space is R0. That is, the Kolmogorov quotient of a T1 space is also T1 (because they are homeomorphic). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre)
Theorem	ist1-5 23709	A topological space is T1 iff it is both T0 and R0. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Fre))
Theorem	ishaus3 23710	A topological space is Hausdorff iff it is both T0 and R1 (where R1 means that any two topologically distinct points are separated by neighborhoods). (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Kol2 ∧ (KQ‘𝐽) ∈ Haus))
Theorem	nrmreg 23711	A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 23636. (Contributed by Mario Carneiro, 25-Aug-2015.)
⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg)
Theorem	reghaus 23712	A regular T0 space is Hausdorff. In other words, a T3 space is T2 . A regular Hausdorff or T0 space is also known as a T3 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐽 ∈ Reg → (𝐽 ∈ Haus ↔ 𝐽 ∈ Kol2))
Theorem	nrmhaus 23713	A T1 normal space is Hausdorff. A Hausdorff or T1 normal space is also known as a T4 space. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ (𝐽 ∈ Nrm → (𝐽 ∈ Haus ↔ 𝐽 ∈ Fre))
12.2  Filters and filter bases
12.2.1  Filter bases
Theorem	elmptrab 23714*	Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓))
Theorem	elmptrab2 23715*	Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015.) (Revised by AV, 26-Mar-2021.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)    &   ⊢ 𝐵 ∈ V    &   ⊢ (𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓))
Theorem	isfbas 23716*	The predicate "𝐹 is a filter base." Note that some authors require filter bases to be closed under pairwise intersections, but that is not necessary under our definition. One advantage of this definition is that tails in a directed set form a filter base under our meaning. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 (𝐹 ∩ 𝒫 (𝑥 ∩ 𝑦)) ≠ ∅))))
Theorem	fbasne0 23717	There are no empty filter bases. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ≠ ∅)
Theorem	0nelfb 23718	No filter base contains the empty set. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Mario Carneiro, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → ¬ ∅ ∈ 𝐹)
Theorem	fbsspw 23719	A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐹 ⊆ 𝒫 𝐵)
Theorem	fbelss 23720	An element of the filter base is a subset of the base set. (Contributed by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝑋 ∈ 𝐹) → 𝑋 ⊆ 𝐵)
Theorem	fbdmn0 23721	The domain of a filter base is nonempty. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐹 ∈ (fBas‘𝐵) → 𝐵 ≠ ∅)
Theorem	isfbas2 23722*	The predicate "𝐹 is a filter base." (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ (𝐵 ∈ 𝐴 → (𝐹 ∈ (fBas‘𝐵) ↔ (𝐹 ⊆ 𝒫 𝐵 ∧ (𝐹 ≠ ∅ ∧ ∅ ∉ 𝐹 ∧ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐹 ∃𝑧 ∈ 𝐹 𝑧 ⊆ (𝑥 ∩ 𝑦)))))
Theorem	fbasssin 23723*	A filter base contains subsets of its pairwise intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Jeff Hankins, 1-Dec-2010.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ (𝐴 ∩ 𝐵))
Theorem	fbssfi 23724*	A filter base contains subsets of its finite intersections. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ (fi‘𝐹)) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝐴)
Theorem	fbssint 23725*	A filter base contains subsets of its finite intersections. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝐵) ∧ 𝐴 ⊆ 𝐹 ∧ 𝐴 ∈ Fin) → ∃𝑥 ∈ 𝐹 𝑥 ⊆ ∩ 𝐴)
Theorem	fbncp 23726	A filter base does not contain complements of its elements. (Contributed by Mario Carneiro, 26-Nov-2013.) (Revised by Stefan O'Rear, 28-Jul-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐴 ∈ 𝐹) → ¬ (𝐵 ∖ 𝐴) ∈ 𝐹)
Theorem	fbun 23727*	A necessary and sufficient condition for the union of two filter bases to also be a filter base. (Contributed by Mario Carneiro, 28-Nov-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝐺 ∈ (fBas‘𝑋)) → ((𝐹 ∪ 𝐺) ∈ (fBas‘𝑋) ↔ ∀𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝐺 ∃𝑧 ∈ (𝐹 ∪ 𝐺)𝑧 ⊆ (𝑥 ∩ 𝑦)))
Theorem	fbfinnfr 23728	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 23729*	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 23730	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 23731*	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 23732	Extend class notation with the set of filters on a set.
class Fil
Definition	df-fil 23733*	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 23734*	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 23735	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 23736	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 23737	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 23738	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 23739	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 23740	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 23741	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 23742	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 23743*	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 23744*	Lemma for isfild 23745. (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 23745*	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 23746	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 23747	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 23748	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 23749	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 23750	The intersection of two filters is a filter. Use fiint 9277 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 23751	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 23752	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 23753	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 23754	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 23755	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 23756*	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 23757*	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 23758*	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 23759	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 23760	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 23761*	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 23762	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 23763*	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 23764	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 23765	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 23766	Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹))
Theorem	neifil 23767	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 23768	Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋)
Theorem	filunirn 23769	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 23770	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 23771*	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 23772*	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 23773	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 23774*	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 23775	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 23776	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 23777	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 23778	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 23777 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 23779	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 23774 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 23780*	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 23781*	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 23782*	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 23783	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 23784	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 23785	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 23786	Extend class notation with the ultrafilters-on-a-set function.
class UFil
Syntax	cufl 23787	Extend class notation with the ultrafilter lemma.
class UFL
Definition	df-ufil 23788*	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 23789*	Define the class of base sets for which the ultrafilter lemma filssufil 23799 holds. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔}
Theorem	isufil 23790*	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 23791	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 23792	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 23793	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 23794	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 23795*	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 23796	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 23797	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 23798*	A filter is contained in some ultrafilter. This version of filssufil 23799 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 23799*	A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 10423.) (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 23800*	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))