P. 240 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fmf 23901	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 23902	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 23903*	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 23904*	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 23905	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 23906*	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 23907	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 23908*	Lemma for rnelfm 23909. (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 23909	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 23910*	Lemma for fmfnfm 23914. (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 23911*	Lemma for fmfnfm 23914. (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 23912*	Lemma for fmfnfm 23914. (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 23913*	Lemma for fmfnfm 23914. (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 23914*	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 23915	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 23916	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 23917	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 23918	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 23919*	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 23920	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 23921	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 23922	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 23923	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 23924	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 23925	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 23926	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 23927	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 23928	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 23929	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 23930	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 23931*	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 23932*	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 23933*	A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 23932. (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)))
Theorem	flimclsi 23934	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.)
⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Theorem	hausflimlem 23935	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)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑈 ∩ 𝑉) ≠ ∅)
Theorem	hausflimi 23936*	One direction of hausflim 23937. 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))
Theorem	hausflim 23937*	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)))
Theorem	flimcf 23938*	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)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)))
Theorem	flimrest 23939	The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))
Theorem	flimclslem 23940	Lemma for flimcls 23941. (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) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹)))
Theorem	flimcls 23941*	Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑆 ∈ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓))))
Theorem	flimsncls 23942	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))
Theorem	hauspwpwf1 23943*	Lemma for hauspwpwdom 23944. 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 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))})    ⇒   ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴)
Theorem	hauspwpwdom 23944	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 ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴)
Theorem	flffval 23945*	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 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
Theorem	flfval 23946	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:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
Theorem	flfnei 23947*	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:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛)))
Theorem	flfneii 23948*	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:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑁)
Theorem	isflf 23949*	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:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	flfelbas 23950	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:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → 𝐴 ∈ 𝑋)
Theorem	flffbas 23951*	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:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	flftg 23952*	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:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜))))
Theorem	hausflf 23953*	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:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹))
Theorem	hausflf2 23954	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:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o)
Theorem	cnpflfi 23955	Forward direction of cnpflf 23957. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
Theorem	cnpflf2 23956	𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹‘𝐴). 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) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))
Theorem	cnpflf 23957*	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) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))))
Theorem	cnflf 23958*	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:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))
Theorem	cnflf2 23959*	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:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐹 “ (𝐽 fLim 𝑓)) ⊆ ((𝐾 fLimf 𝑓)‘𝐹))))
Theorem	flfcnp 23960	A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹)))
Theorem	lmflf 23961	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 = (ℤ≥‘𝑀)    &   ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍))    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹)))
Theorem	txflf 23962*	Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))    &   ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)    &   ⊢ (𝜑 → 𝐺:𝑍⟶𝑌)    &   ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ ⟨(𝐹‘𝑛), (𝐺‘𝑛)⟩)    ⇒   ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺))))
Theorem	flfcnp2 23963*	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.)
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴)))    &   ⊢ (𝜑 → 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵)))    &   ⊢ (𝜑 → 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))    ⇒   ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
Theorem	fclsval 23964*	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, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅))
Theorem	isfcls 23965*	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) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Theorem	fclsfil 23966	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))
Theorem	fclstop 23967	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)
Theorem	fclstopon 23968	Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋)))
Theorem	isfcls2 23969*	A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))
Theorem	fclsopn 23970*	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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))))
Theorem	fclsopni 23971	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) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅)
Theorem	fclselbas 23972	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) → 𝐴 ∈ 𝑋)
Theorem	fclsneii 23973	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}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅)
Theorem	fclssscls 23974	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.)
⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆))
Theorem	fclsnei 23975*	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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅)))
Theorem	supnfcls 23976*	The filter of supersets of 𝑋 ∖ 𝑈 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) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑈) ⊆ 𝑥}))
Theorem	fclsbas 23977*	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) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅))))
Theorem	fclsss1 23978	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) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹))
Theorem	fclsss2 23979	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) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹))
Theorem	fclsrest 23980	The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fClus (𝐹 ↾t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌))
Theorem	fclscf 23981*	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)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)))
Theorem	flimfcls 23982	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)
Theorem	fclsfnflim 23983*	A filter clusters at a point iff a finer filter converges to it. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∃𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 ∧ 𝐴 ∈ (𝐽 fLim 𝑔))))
Theorem	flimfnfcls 23984*	A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 23983, this theorem illustrates the duality between convergence and clustering. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹 ⊆ 𝑔 → 𝐴 ∈ (𝐽 fClus 𝑔))))
Theorem	fclscmpi 23985	Forward direction of fclscmp 23986. Every filter clusters in a compact space. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Comp ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐽 fClus 𝐹) ≠ ∅)
Theorem	fclscmp 23986*	A space is compact iff every filter clusters. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅))
Theorem	uffclsflim 23987	The cluster points of an ultrafilter are its limit points. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
Theorem	ufilcmp 23988*	A space is compact iff every ultrafilter converges. (Contributed by Jeff Hankins, 11-Dec-2009.) (Proof shortened by Mario Carneiro, 12-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.)
⊢ ((𝑋 ∈ UFL ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (UFil‘𝑋)(𝐽 fLim 𝑓) ≠ ∅))
Theorem	fcfval 23989	The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
Theorem	isfcf 23990*	The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))
Theorem	fcfnei 23991*	The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
Theorem	fcfelbas 23992	A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹)) → 𝐴 ∈ 𝑋)
Theorem	fcfneii 23993	A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐿)) → (𝑁 ∩ (𝐹 “ 𝑆)) ≠ ∅)
Theorem	flfssfcf 23994	A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) ⊆ ((𝐽 fClusf 𝐿)‘𝐹))
Theorem	uffcfflf 23995	If the domain filter is an ultrafilter, the cluster points of the function are the limit points. (Contributed by Jeff Hankins, 12-Dec-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = ((𝐽 fLimf 𝐿)‘𝐹))
Theorem	cnpfcfi 23996	Lemma for cnpfcf 23997. If a function is continuous at a point, it respects clustering there. (Contributed by Jeff Hankins, 20-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐾 ∈ Top ∧ 𝐴 ∈ (𝐽 fClus 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))
Theorem	cnpfcf 23997*	A function 𝐹 is continuous at point 𝐴 iff 𝐹 respects cluster points there. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))
Theorem	cnfcf 23998*	Continuity of a function in terms of cluster points of a function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))
Theorem	flfcntr 23999	A continuous function's value is always in the trace of its filter limit. (Contributed by Thierry Arnoux, 30-Aug-2020.)
⊢ 𝐶 = ∪ 𝐽    &   ⊢ 𝐵 = ∪ 𝐾    &   ⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐶)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))
Theorem	alexsublem 24000*	Lemma for alexsub 24001. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝜑 → 𝑋 ∈ UFL)    &   ⊢ (𝜑 → 𝑋 = ∪ 𝐵)    &   ⊢ (𝜑 → 𝐽 = (topGen‘(fi‘𝐵)))    &   ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)    &   ⊢ (𝜑 → 𝐹 ∈ (UFil‘𝑋))    &   ⊢ (𝜑 → (𝐽 fLim 𝐹) = ∅)    ⇒   ⊢ ¬ 𝜑