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

Theoremisufil 21701* 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 21702 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 21703 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 21704 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 21705 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 21706* 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 21707 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 21708 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 21709* A filter is contained in some ultrafilter. This version of filssufil 21710 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 21710* A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 9289.) (Contributed by Jeff Hankins, 2-Dec-2009.) (Revised by Stefan O'Rear, 29-Jul-2015.)
(𝐹 ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)

Theoremisufl 21711* 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 21712* 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 21713 Consequence of filssufilg 21709: 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 21714 A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑋 ∈ Fin → 𝑋 ∈ UFL)

Theoremacufl 21715 The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.)
(CHOICE → UFL = V)

Theoremssufl 21716 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 21717* 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 21718* 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 21719* Lemma for fixufil 21720 and uffixfr 21721. (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 21720* 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 21721* 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 21722* 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 21723 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 21724 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 21725* 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 21726* An ultrafilter is free iff it contains the Fréchet filter cfinfil 21691 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 21727* 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 21728* 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 21729 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 21730 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 21731 Extend class definition to include the neighborhood filter mapping function.
class FilMap

Syntaxcflim 21732 Extend class notation with a function returning the limit of a filter.
class fLim

Syntaxcflf 21733 Extend class definition to include the function for filter-based function limits.
class fLimf

Syntaxcfcls 21734 Extend class definition to include the cluster point function on filters.
class fClus

Syntaxcfcf 21735 Extend class definition to include the function for cluster points of a function.
class fClusf

Definitiondf-fm 21736* 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 21737* 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 21738* 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 21739* 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 21740* 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 21741* 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 21742 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 21743 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 21744 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 21745* 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 21746* 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 21747 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 21748* 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 21749 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 21750* Lemma for rnelfm 21751. (Contributed by Jeff Hankins, 14-Nov-2009.)
(((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))

Theoremrnelfm 21751 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 21752* Lemma for fmfnfm 21756. (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 21753* Lemma for fmfnfm 21756. (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 21754* Lemma for fmfnfm 21756. (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 21755* Lemma for fmfnfm 21756. (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 21756* 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 21757 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 21758 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 21759 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 21760 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 21761* 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 21762 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 21763 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 21764 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 21765 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 21766 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 21767 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 21768 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 21769 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 21770 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 21771 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 21772 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 21773* 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 21774* 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 21775* A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 21774. (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 21776 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 21777 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 21778* One direction of hausflim 21779. 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 21779* 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 21780* 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 21781 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 21782 Lemma for flimcls 21783. (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 21783* 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 21784 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 21785* Lemma for hauspwpwdom 21786. 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 21786 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 21787* 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 21788 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 21789* 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 21790* 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 21791* 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 21792 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 21793* 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 21794* 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 21795* 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 21796 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 21797 Forward direction of cnpflf 21799. (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 21798 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 21799* 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 21800* 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))))

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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