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Theorem List for Metamath Proof Explorer - 21801-21900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelfm3 21801* 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 21802 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 21803* Lemma for rnelfm 21804. (Contributed by Jeff Hankins, 14-Nov-2009.)
(((𝑌𝐴𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌𝑋) ∧ ran 𝐹𝐿) → ran (𝑥𝐿 ↦ (𝐹𝑥)) ∈ (fBas‘𝑌))

Theoremrnelfm 21804 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 21805* Lemma for fmfnfm 21809. (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 21806* Lemma for fmfnfm 21809. (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 21807* Lemma for fmfnfm 21809. (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 21808* Lemma for fmfnfm 21809. (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 21809* 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 21810 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 21811 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 21812 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 21813 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 21814* 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 21815 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 21816 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 21817 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 21818 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 21819 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 21820 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 21821 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 21822 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 21823 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 21824 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 21825 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 21826* 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 21827* 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 21828* A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 21827. (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 21829 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 21830 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 21831* One direction of hausflim 21832. 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 21832* 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 21833* 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 21834 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 21835 Lemma for flimcls 21836. (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 21836* 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 21837 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 21838* Lemma for hauspwpwdom 21839. 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 21839 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 21840* 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 21841 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 21842* 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 21843* 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 21844* 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 21845 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 21846* 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 21847* 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 21848* 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 21849 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 21850 Forward direction of cnpflf 21852. (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 21851 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 21852* 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 21853* 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 21854* 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 21855 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 21856 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 21857* 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 21858* 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 21859* 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 21860* 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 21861 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 21862 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 21863 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 21864* 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 21865* 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 21866 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 21867 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 21868 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 21869 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 21870* 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 21871* 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 21872* 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 21873 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 21874 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 21875 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 21876* 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 21877 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)

Theoremfclsfnflim 21878* 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)(𝐹𝑔𝐴 ∈ (𝐽 fLim 𝑔))))

Theoremflimfnfcls 21879* A filter converges to a point iff every finer filter clusters there. Along with fclsfnflim 21878, 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)(𝐹𝑔𝐴 ∈ (𝐽 fClus 𝑔))))

Theoremfclscmpi 21880 Forward direction of fclscmp 21881. 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) ≠ ∅)

Theoremfclscmp 21881* 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) ≠ ∅))

Theoremuffclsflim 21882 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))

Theoremufilcmp 21883* 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) ≠ ∅))

Theoremfcfval 21884 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:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))

Theoremisfcf 21885* 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:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))

Theoremfcfnei 21886* 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:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠𝐿 (𝑛 ∩ (𝐹𝑠)) ≠ ∅)))

Theoremfcfelbas 21887 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:𝑌𝑋) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹)) → 𝐴𝑋)

Theoremfcfneii 21888 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:𝑌𝑋) ∧ (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆𝐿)) → (𝑁 ∩ (𝐹𝑆)) ≠ ∅)

Theoremflfssfcf 21889 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:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) ⊆ ((𝐽 fClusf 𝐿)‘𝐹))

Theoremuffcfflf 21890 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:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = ((𝐽 fLimf 𝐿)‘𝐹))

Theoremcnpfcfi 21891 Lemma for cnpfcf 21892. 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)) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝐿)‘𝐹))

Theoremcnpfcf 21892* 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) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fClus 𝑓) → (𝐹𝐴) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))))

Theoremcnfcf 21893* 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:𝑋𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))

Theoremflfcntr 21894 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    &   (𝜑𝐽 ∈ Top)    &   (𝜑𝐴𝐶)    &   (𝜑𝐹 ∈ ((𝐽t 𝐴) Cn 𝐾))    &   (𝜑𝑋𝐴)       (𝜑 → (𝐹𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))

Theoremalexsublem 21895* Lemma for alexsub 21896. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝜑𝑋 ∈ UFL)    &   (𝜑𝑋 = 𝐵)    &   (𝜑𝐽 = (topGen‘(fi‘𝐵)))    &   ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)    &   (𝜑𝐹 ∈ (UFil‘𝑋))    &   (𝜑 → (𝐽 fLim 𝐹) = ∅)        ¬ 𝜑

Theoremalexsub 21896* The Alexander Subbase Theorem: If 𝐵 is a subbase for the topology 𝐽, and any cover taken from 𝐵 has a finite subcover, then the generated topology is compact. This proof uses the ultrafilter lemma; see alexsubALT 21902 for a proof using Zorn's lemma. (Contributed by Jeff Hankins, 24-Jan-2010.) (Revised by Mario Carneiro, 26-Aug-2015.)
(𝜑𝑋 ∈ UFL)    &   (𝜑𝑋 = 𝐵)    &   (𝜑𝐽 = (topGen‘(fi‘𝐵)))    &   ((𝜑 ∧ (𝑥𝐵𝑋 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)       (𝜑𝐽 ∈ Comp)

Theoremalexsubb 21897* Biconditional form of the Alexander Subbase Theorem alexsub 21896. (Contributed by Mario Carneiro, 27-Aug-2015.)
((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)))

TheoremalexsubALTlem1 21898* Lemma for alexsubALT 21902. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
𝑋 = 𝐽       (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))

TheoremalexsubALTlem2 21899* Lemma for alexsubALT 21902. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
𝑋 = 𝐽       (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = 𝑏)} ∪ {∅}) ¬ 𝑢𝑣)

TheoremalexsubALTlem3 21900* Lemma for alexsubALT 21902. If a point is covered by a collection taken from the base with no finite subcover, a set from the subbase can be added that covers the point so that the resulting collection has no finite subcover. (Contributed by Jeff Hankins, 28-Jan-2010.) (Revised by Mario Carneiro, 14-Dec-2013.)
𝑋 = 𝐽       (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ (𝑢 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎𝑢 ∧ ∀𝑏 ∈ (𝒫 𝑢 ∩ Fin) ¬ 𝑋 = 𝑏))) ∧ 𝑤𝑢) ∧ ((𝑡 ∈ (𝒫 𝑥 ∩ Fin) ∧ 𝑤 = 𝑡) ∧ (𝑦𝑤 ∧ ¬ 𝑦 (𝑥𝑢)))) → ∃𝑠𝑡𝑛 ∈ (𝒫 (𝑢 ∪ {𝑠}) ∩ Fin) ¬ 𝑋 = 𝑛)

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