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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fgcl 23901 | 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 23902 | Absorption law for filter generation. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝐹 ∈ (fBas‘𝑌) ∧ 𝑌 ⊆ 𝑋) → (𝑋filGen(𝑌filGen𝐹)) = (𝑋filGen𝐹)) | ||
Theorem | neifil 23903 | 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 23904 | Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 = 𝑋) | ||
Theorem | filunirn 23905 | 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 23906 | 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 23907* | 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 23908* | 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 23909 | 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 23910* | 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 23911 | 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 23912 | 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 23913 | 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 23914 | 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 23913 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 23915 | 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 23910 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 23916* | 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 23917* | 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 23918* | 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 23919 | 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 23920 | 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 23921 | 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)) | ||
Syntax | cufil 23922 | Extend class notation with the ultrafilters-on-a-set function. |
class UFil | ||
Syntax | cufl 23923 | Extend class notation with the ultrafilter lemma. |
class UFL | ||
Definition | df-ufil 23924* | 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 23925* | Define the class of base sets for which the ultrafilter lemma filssufil 23935 holds. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ UFL = {𝑥 ∣ ∀𝑓 ∈ (Fil‘𝑥)∃𝑔 ∈ (UFil‘𝑥)𝑓 ⊆ 𝑔} | ||
Theorem | isufil 23926* | 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 23927 | 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 23928 | 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 23929 | 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 23930 | 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 23931* | 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 23932 | 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 23933 | 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 23934* | A filter is contained in some ultrafilter. This version of filssufil 23935 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 23935* | A filter is contained in some ultrafilter. (Requires the Axiom of Choice, via numth3 10507.) (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 23936* | 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 23937* | 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 23938 | Consequence of filssufilg 23934: 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 23939 | A finite set satisfies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝑋 ∈ Fin → 𝑋 ∈ UFL) | ||
Theorem | acufl 23940 | The axiom of choice implies the ultrafilter lemma. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (CHOICE → UFL = V) | ||
Theorem | ssufl 23941 | 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 23942* | 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 23943* | 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 23944* | Lemma for fixufil 23945 and uffixfr 23946. (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 23945* | 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 23946* | 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 23947* | 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 23948 | 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 23949 | 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 23950* | 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 23951* | An ultrafilter is free iff it contains the Fréchet filter cfinfil 23916 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 23952* | 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 23953* | 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 23954 | 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 23955 | 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 = ∅)) | ||
Syntax | cfm 23956 | Extend class definition to include the neighborhood filter mapping function. |
class FilMap | ||
Syntax | cflim 23957 | Extend class notation with a function returning the limit of a filter. |
class fLim | ||
Syntax | cflf 23958 | Extend class definition to include the function for filter-based function limits. |
class fLimf | ||
Syntax | cfcls 23959 | Extend class definition to include the cluster point function on filters. |
class fClus | ||
Syntax | cfcf 23960 | Extend class definition to include the function for cluster points of a function. |
class fClusf | ||
Definition | df-fm 23961* | 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 23962* | 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 23963* | 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)))) | ||
Definition | df-fcls 23964* | 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, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅)) | ||
Definition | df-fcf 23965* | 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 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)))) | ||
Theorem | fmval 23966* | 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.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))) | ||
Theorem | fmfil 23967 | A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋)) | ||
Theorem | fmf 23968 | Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋)) | ||
Theorem | fmss 23969 | A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶)) | ||
Theorem | elfm 23970* | An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴))) | ||
Theorem | elfm2 23971* | An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))) | ||
Theorem | fmfg 23972 | 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) ⇒ ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿)) | ||
Theorem | elfm3 23973* | 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:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) | ||
Theorem | imaelfm 23974 | 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) ⇒ ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → (𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵)) | ||
Theorem | rnelfmlem 23975* | Lemma for rnelfm 23976. (Contributed by Jeff Hankins, 14-Nov-2009.) |
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) | ||
Theorem | rnelfm 23976 | 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.) |
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹 ∈ 𝐿)) | ||
Theorem | fmfnfmlem1 23977* | Lemma for fmfnfm 23981. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) | ||
Theorem | fmfnfmlem2 23978* | Lemma for fmfnfm 23981. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) | ||
Theorem | fmfnfmlem3 23979* | Lemma for fmfnfm 23981. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) | ||
Theorem | fmfnfmlem4 23980* | Lemma for fmfnfm 23981. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡))) | ||
Theorem | fmfnfm 23981* | 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.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ (Fil‘𝑌)(𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓))) | ||
Theorem | fmufil 23982 | An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋)) | ||
Theorem | fmid 23983 | 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) | ||
Theorem | fmco 23984 | Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))) | ||
Theorem | ufldom 23985 | 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 ∧ 𝑌 ≼ 𝑋) → 𝑌 ∈ UFL) | ||
Theorem | flimval 23986* | 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) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}) | ||
Theorem | elflim2 23987 | 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) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) | ||
Theorem | flimtop 23988 | 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) | ||
Theorem | flimneiss 23989 | 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) | ||
Theorem | flimnei 23990 | 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})) → 𝑁 ∈ 𝐹) | ||
Theorem | flimelbas 23991 | 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) → 𝐴 ∈ 𝑋) | ||
Theorem | flimfil 23992 | 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)) | ||
Theorem | flimtopon 23993 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) | ||
Theorem | elflim 23994 | 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) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) | ||
Theorem | flimss2 23995 | 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) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹)) | ||
Theorem | flimss1 23996 | 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) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹)) | ||
Theorem | neiflim 23997 | 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) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))) | ||
Theorem | flimopn 23998* | 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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))) | ||
Theorem | fbflim 23999* | 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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))) | ||
Theorem | fbflim2 24000* | A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23999. (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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))) |
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