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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuffinfix 22101* 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 22102* An ultrafilter is free iff it contains the Fréchet filter cfinfil 22067 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 22103* 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 22104* 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 22105 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 22106 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 22107 Extend class definition to include the neighborhood filter mapping function.
class FilMap

Syntaxcflim 22108 Extend class notation with a function returning the limit of a filter.
class fLim

Syntaxcflf 22109 Extend class definition to include the function for filter-based function limits.
class fLimf

Syntaxcfcls 22110 Extend class definition to include the cluster point function on filters.
class fClus

Syntaxcfcf 22111 Extend class definition to include the function for cluster points of a function.
class fClusf

Definitiondf-fm 22112* 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 22113* 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 22114* 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 22115* 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 22116* 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 22117* 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 22118 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 22119 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 22120 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 22121* 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 22122* 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 22123 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))

Theoremelfm3 22124* An alternate formulation of elementhood in a mapping filter that requires 𝐹 to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐿 = (𝑌filGen𝐵)       ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌onto𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥𝐿 𝐴 = (𝐹𝑥)))

Theoremimaelfm 22125 An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐿 = (𝑌filGen𝐵)       (((𝑋𝐴𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑆𝐿) → (𝐹𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵))

Theoremrnelfmlem 22126* Lemma for rnelfm 22127. (Contributed by Jeff Hankins, 14-Nov-2009.)
(((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))

Theoremrnelfm 22127 A condition for a filter to be an image filter for a given function. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹𝐿))

Theoremfmfnfmlem1 22128* Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝜑𝐵 ∈ (fBas‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑋))    &   (𝜑𝐹:𝑌𝑋)    &   (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)       (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))

Theoremfmfnfmlem2 22129* Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝜑𝐵 ∈ (fBas‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑋))    &   (𝜑𝐹:𝑌𝑋)    &   (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)       (𝜑 → (∃𝑥𝐿 𝑠 = (𝐹𝑥) → ((𝐹𝑠) ⊆ 𝑡 → (𝑡𝑋𝑡𝐿))))

Theoremfmfnfmlem3 22130* Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝜑𝐵 ∈ (fBas‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑋))    &   (𝜑𝐹:𝑌𝑋)    &   (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)       (𝜑 → (fi‘ran (𝑥𝐿 ↦ (𝐹𝑥))) = ran (𝑥𝐿 ↦ (𝐹𝑥)))

Theoremfmfnfmlem4 22131* Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝜑𝐵 ∈ (fBas‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑋))    &   (𝜑𝐹:𝑌𝑋)    &   (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)       (𝜑 → (𝑡𝐿 ↔ (𝑡𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥𝐿 ↦ (𝐹𝑥))))(𝐹𝑠) ⊆ 𝑡)))

Theoremfmfnfm 22132* A filter finer than an image filter is an image filter of the same function. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝜑𝐵 ∈ (fBas‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑋))    &   (𝜑𝐹:𝑌𝑋)    &   (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿)       (𝜑 → ∃𝑓 ∈ (Fil‘𝑌)(𝐵𝑓𝐿 = ((𝑋 FilMap 𝐹)‘𝑓)))

Theoremfmufil 22133 An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝑋𝐴𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋))

Theoremfmid 22134 The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)
(𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)

Theoremfmco 22135 Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.)
(((𝑋𝑉𝑌𝑊𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌𝑋𝐺:𝑍𝑌)) → ((𝑋 FilMap (𝐹𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵)))

Theoremufldom 22136 The ultrafilter lemma property is a cardinal invariant, so since it transfers to subsets it also transfers over set dominance. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝑋 ∈ UFL ∧ 𝑌𝑋) → 𝑌 ∈ UFL)

Theoremflimval 22137* The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐹 ran Fil) → (𝐽 fLim 𝐹) = {𝑥𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹𝐹 ⊆ 𝒫 𝑋)})

Theoremelflim2 22138 The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Theoremflimtop 22139 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
(𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top)

Theoremflimneiss 22140 A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
(𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)

Theoremflimnei 22141 A filter contains all of the neighborhoods of its limit points. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁𝐹)

Theoremflimelbas 22142 A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴𝑋)

Theoremflimfil 22143 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Theoremflimtopon 22144 Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Theoremelflim 22145 The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))

Theoremflimss2 22146 A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹))

Theoremflimss1 22147 A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Theoremneiflim 22148 A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))

Theoremflimopn 22149* The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥𝑥𝐹))))

Theoremfbflim 22150* A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐹 = (𝑋filGen𝐵)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑥𝐽 (𝐴𝑥 → ∃𝑦𝐵 𝑦𝑥))))

Theoremfbflim2 22151* A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 22150. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐹 = (𝑋filGen𝐵)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥𝐵 𝑥𝑛)))

Theoremflimclsi 22152 The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
(𝑆𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Theoremhausflimlem 22153 If 𝐴 and 𝐵 are both limits of the same filter, then all neighborhoods of 𝐴 and 𝐵 intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
(((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈𝐽𝑉𝐽) ∧ (𝐴𝑈𝐵𝑉)) → (𝑈𝑉) ≠ ∅)

Theoremhausflimi 22154* One direction of hausflim 22155. A filter in a Hausdorff space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.)
(𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹))

Theoremhausflim 22155* A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))

Theoremflimcf 22156* Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)))

Theoremflimrest 22157 The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fLim (𝐹t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))

Theoremflimclslem 22158 Lemma for flimcls 22159. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝐴 ∈ (𝐽 fLim 𝐹)))

Theoremflimcls 22159* Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → (𝐴 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑆𝑓𝐴 ∈ (𝐽 fLim 𝑓))))

Theoremflimsncls 22160 If 𝐴 is a limit point of the filter 𝐹, then all the points which specialize 𝐴 (in the specialization preorder) are also limit points. Thus, the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.)
(𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹))

Theoremhauspwpwf1 22161* Lemma for hauspwpwdom 22162. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.)
𝑋 = 𝐽    &   𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗𝐽 (𝑥𝑗𝑎 = (𝑗𝐴))})       ((𝐽 ∈ Haus ∧ 𝐴𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)

Theoremhauspwpwdom 22162 If 𝑋 is a Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by the double powerset of 𝐴. In particular, a Hausdorff space with a dense subset 𝐴 has cardinality at most 𝒫 𝒫 𝐴, and a separable Hausdorff space has cardinality at most 𝒫 𝒫 ℕ. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)

Theoremflffval 22163* Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))

Theoremflfval 22164 Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))

Theoremflfnei 22165* The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))

Theoremflfneii 22166* A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑁)

Theoremisflf 22167* The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))

Theoremflfelbas 22168 A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → 𝐴𝑋)

Theoremflffbas 22169* Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
𝐿 = (𝑌filGen𝐵)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))

Theoremflftg 22170* Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.)
𝐽 = (topGen‘𝐵)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐵 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))

Theoremhausflf 22171* If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))

Theoremhausflf2 22172 If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.)
𝑋 = 𝐽       (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)

Theoremcnpflfi 22173 Forward direction of cnpflf 22175. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))

Theoremcnpflf2 22174 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹𝐴). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
𝐿 = ((nei‘𝐽)‘{𝐴})       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))

Theoremcnpflf 22175* Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))

Theoremcnflf 22176* A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))

Theoremcnflf2 22177* A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐹 “ (𝐽 fLim 𝑓)) ⊆ ((𝐾 fLimf 𝑓)‘𝐹))))

Theoremflfcnp 22178 A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
(((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺𝐹)))

Theoremlmflf 22179 The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐿 = (𝑍filGen(ℤ𝑍))       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍𝑋) → (𝐹(⇝𝑡𝐽)𝑃𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))

Theoremtxflf 22180* Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑍))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑𝐺:𝑍𝑌)    &   𝐻 = (𝑛𝑍 ↦ ⟨(𝐹𝑛), (𝐺𝑛)⟩)       (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))

Theoremflfcnp2 22181* The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)
(𝜑𝐽 ∈ (TopOn‘𝑋))    &   (𝜑𝐾 ∈ (TopOn‘𝑌))    &   (𝜑𝐿 ∈ (Fil‘𝑍))    &   ((𝜑𝑥𝑍) → 𝐴𝑋)    &   ((𝜑𝑥𝑍) → 𝐵𝑌)    &   (𝜑𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥𝑍𝐴)))    &   (𝜑𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥𝑍𝐵)))    &   (𝜑𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))       (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥𝑍 ↦ (𝐴𝑂𝐵))))

Theoremfclsval 22182* The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, 𝑡𝐹 ((cls‘𝐽)‘𝑡), ∅))

Theoremisfcls 22183* A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Theoremfclsfil 22184 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋))

Theoremfclstop 22185 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top)

Theoremfclstopon 22186 Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))

Theoremisfcls2 22187* A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))

Theoremfclsopn 22188* Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐹 (𝑜𝑠) ≠ ∅))))

Theoremfclsopni 22189 An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈𝐽𝐴𝑈𝑆𝐹)) → (𝑈𝑆) ≠ ∅)

Theoremfclselbas 22190 A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
𝑋 = 𝐽       (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴𝑋)

Theoremfclsneii 22191 A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐹) → (𝑁𝑆) ≠ ∅)

Theoremfclssscls 22192 The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝑆𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))

Theoremfclsnei 22193* Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐹 (𝑛𝑠) ≠ ∅)))

Theoremsupnfcls 22194* The filter of supersets of 𝑋𝑈 does not cluster at any point of the open set 𝑈. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈𝐽𝐴𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑈) ⊆ 𝑥}))

Theoremfclsbas 22195* Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
𝐹 = (𝑋filGen𝐵)       ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐵 (𝑜𝑠) ≠ ∅))))

Theoremfclsss1 22196 A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))

Theoremfclsss2 22197 A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))

Theoremfclsrest 22198 The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → ((𝐽t 𝑌) fClus (𝐹t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))

Theoremfclscf 22199* Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))

Theoremflimfcls 22200 A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
(𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹)

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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