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Theorem List for Metamath Proof Explorer - 35701-35800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrankpwg 35701 The rank of a power set. Closed form of rankpw 9858. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ 𝑉 β†’ (rankβ€˜π’« 𝐴) = suc (rankβ€˜π΄))
 
Theoremrank0 35702 The rank of the empty set is βˆ…. (Contributed by Scott Fenton, 17-Jul-2015.)
(rankβ€˜βˆ…) = βˆ…
 
Theoremrankeq1o 35703 The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
((rankβ€˜π΄) = 1o ↔ 𝐴 = {βˆ…})
 
21.10.21  Hereditarily Finite Sets
 
Syntaxchf 35704 The constant Hf is a class.
class Hf
 
Definitiondf-hf 35705 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
Hf = βˆͺ (𝑅1 β€œ Ο‰)
 
Theoremelhf 35706* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
(𝐴 ∈ Hf ↔ βˆƒπ‘₯ ∈ Ο‰ 𝐴 ∈ (𝑅1β€˜π‘₯))
 
Theoremelhf2 35707 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
𝐴 ∈ V    β‡’   (𝐴 ∈ Hf ↔ (rankβ€˜π΄) ∈ Ο‰)
 
Theoremelhf2g 35708 Hereditarily finiteness via rank. Closed form of elhf2 35707. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴 ∈ 𝑉 β†’ (𝐴 ∈ Hf ↔ (rankβ€˜π΄) ∈ Ο‰))
 
Theorem0hf 35709 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
βˆ… ∈ Hf
 
Theoremhfun 35710 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐡 ∈ Hf ) β†’ (𝐴 βˆͺ 𝐡) ∈ Hf )
 
Theoremhfsn 35711 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴 ∈ Hf β†’ {𝐴} ∈ Hf )
 
Theoremhfadj 35712 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐡 ∈ Hf ) β†’ (𝐴 βˆͺ {𝐡}) ∈ Hf )
 
Theoremhfelhf 35713 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴 ∈ 𝐡 ∧ 𝐡 ∈ Hf ) β†’ 𝐴 ∈ Hf )
 
Theoremhftr 35714 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
Tr Hf
 
Theoremhfext 35715* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐡 ∈ Hf ) β†’ (𝐴 = 𝐡 ↔ βˆ€π‘₯ ∈ Hf (π‘₯ ∈ 𝐴 ↔ π‘₯ ∈ 𝐡)))
 
Theoremhfuni 35716 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf β†’ βˆͺ 𝐴 ∈ Hf )
 
Theoremhfpw 35717 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf β†’ 𝒫 𝐴 ∈ Hf )
 
Theoremhfninf 35718 Ο‰ is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
Β¬ Ο‰ ∈ Hf
 
21.11  Mathbox for Gino Giotto
 
21.11.1  Study of ax-mulf usage.
 
Theoremmpomulnzcnf 35719* Multiplication maps nonzero complex numbers to nonzero complex numbers. Version of mulnzcnf 11882 using maps-to notation, which does not require ax-mulf 11210. (Contributed by GG, 18-Apr-2025.)
(π‘₯ ∈ (β„‚ βˆ– {0}), 𝑦 ∈ (β„‚ βˆ– {0}) ↦ (π‘₯ Β· 𝑦)):((β„‚ βˆ– {0}) Γ— (β„‚ βˆ– {0}))⟢(β„‚ βˆ– {0})
 
21.12  Mathbox for Jeff Hankins
 
21.12.1  Miscellany
 
Theorema1i14 35720 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)
(πœ“ β†’ (πœ’ β†’ 𝜏))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ 𝜏))))
 
Theorema1i24 35721 Add two antecedents to a wff. Deduction associated with a1i13 27. (Contributed by Jeff Hankins, 5-Aug-2009.)
(πœ‘ β†’ (πœ’ β†’ 𝜏))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ 𝜏))))
 
Theoremexp5d 35722 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((πœ‘ ∧ πœ“) ∧ πœ’) β†’ ((πœƒ ∧ 𝜏) β†’ πœ‚))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp5g 35723 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((πœ‘ ∧ πœ“) β†’ (((πœ’ ∧ πœƒ) ∧ 𝜏) β†’ πœ‚))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp5k 35724 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(πœ‘ β†’ (((πœ“ ∧ (πœ’ ∧ πœƒ)) ∧ 𝜏) β†’ πœ‚))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp56 35725 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((((πœ‘ ∧ πœ“) ∧ πœ’) ∧ (πœƒ ∧ 𝜏)) β†’ πœ‚)    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp58 35726 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
(((πœ‘ ∧ πœ“) ∧ ((πœ’ ∧ πœƒ) ∧ 𝜏)) β†’ πœ‚)    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp510 35727 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((πœ‘ ∧ (((πœ“ ∧ πœ’) ∧ πœƒ) ∧ 𝜏)) β†’ πœ‚)    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp511 35728 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((πœ‘ ∧ ((πœ“ ∧ (πœ’ ∧ πœƒ)) ∧ 𝜏)) β†’ πœ‚)    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theoremexp512 35729 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)
((πœ‘ ∧ ((πœ“ ∧ πœ’) ∧ (πœƒ ∧ 𝜏))) β†’ πœ‚)    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))
 
Theorem3com12d 35730 Commutation in consequent. Swap 1st and 2nd. (Contributed by Jeff Hankins, 17-Nov-2009.)
(πœ‘ β†’ (πœ“ ∧ πœ’ ∧ πœƒ))    β‡’   (πœ‘ β†’ (πœ’ ∧ πœ“ ∧ πœƒ))
 
Theoremimp5p 35731 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))    β‡’   (πœ‘ β†’ (πœ“ β†’ ((πœ’ ∧ πœƒ ∧ 𝜏) β†’ πœ‚)))
 
Theoremimp5q 35732 A triple importation inference. (Contributed by Jeff Hankins, 8-Jul-2009.)
(πœ‘ β†’ (πœ“ β†’ (πœ’ β†’ (πœƒ β†’ (𝜏 β†’ πœ‚)))))    β‡’   ((πœ‘ ∧ πœ“) β†’ ((πœ’ ∧ πœƒ ∧ 𝜏) β†’ πœ‚))
 
Theoremecase13d 35733 Deduction for elimination by cases. (Contributed by Jeff Hankins, 18-Aug-2009.)
(πœ‘ β†’ Β¬ πœ’)    &   (πœ‘ β†’ Β¬ πœƒ)    &   (πœ‘ β†’ (πœ’ ∨ πœ“ ∨ πœƒ))    β‡’   (πœ‘ β†’ πœ“)
 
Theoremsubtr 35734 Transitivity of implicit substitution. (Contributed by Jeff Hankins, 13-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯π‘Œ    &   β„²π‘₯𝑍    &   (π‘₯ = 𝐴 β†’ 𝑋 = π‘Œ)    &   (π‘₯ = 𝐡 β†’ 𝑋 = 𝑍)    β‡’   ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ (𝐴 = 𝐡 β†’ π‘Œ = 𝑍))
 
Theoremsubtr2 35735 Transitivity of implicit substitution into a wff. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
β„²π‘₯𝐴    &   β„²π‘₯𝐡    &   β„²π‘₯πœ“    &   β„²π‘₯πœ’    &   (π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“))    &   (π‘₯ = 𝐡 β†’ (πœ‘ ↔ πœ’))    β‡’   ((𝐴 ∈ 𝐢 ∧ 𝐡 ∈ 𝐷) β†’ (𝐴 = 𝐡 β†’ (πœ“ ↔ πœ’)))
 
Theoremtrer 35736* A relation intersected with its converse is an equivalence relation if the relation is transitive. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
(βˆ€π‘Žβˆ€π‘βˆ€π‘((π‘Ž ≀ 𝑏 ∧ 𝑏 ≀ 𝑐) β†’ π‘Ž ≀ 𝑐) β†’ ( ≀ ∩ β—‘ ≀ ) Er dom ( ≀ ∩ β—‘ ≀ ))
 
Theoremelicc3 35737 An equivalent membership condition for closed intervals. (Contributed by Jeff Hankins, 14-Jul-2009.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐢 ∈ (𝐴[,]𝐡) ↔ (𝐢 ∈ ℝ* ∧ 𝐴 ≀ 𝐡 ∧ (𝐢 = 𝐴 ∨ (𝐴 < 𝐢 ∧ 𝐢 < 𝐡) ∨ 𝐢 = 𝐡))))
 
Theoremfinminlem 35738* A useful lemma about finite sets. If a property holds for a finite set, it holds for a minimal set. (Contributed by Jeff Hankins, 4-Dec-2009.)
(π‘₯ = 𝑦 β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆƒπ‘₯ ∈ Fin πœ‘ β†’ βˆƒπ‘₯(πœ‘ ∧ βˆ€π‘¦((𝑦 βŠ† π‘₯ ∧ πœ“) β†’ π‘₯ = 𝑦)))
 
Theoremgtinf 35739* Any number greater than an infimum is greater than some element of the set. (Contributed by Jeff Hankins, 29-Sep-2013.) (Revised by AV, 10-Oct-2021.)
(((𝑆 βŠ† ℝ ∧ 𝑆 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑆 π‘₯ ≀ 𝑦) ∧ (𝐴 ∈ ℝ ∧ inf(𝑆, ℝ, < ) < 𝐴)) β†’ βˆƒπ‘§ ∈ 𝑆 𝑧 < 𝐴)
 
Theoremopnrebl 35740* A set is open in the standard topology of the reals precisely when every point can be enclosed in an open ball. (Contributed by Jeff Hankins, 23-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGenβ€˜ran (,)) ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦ ∈ ℝ+ ((π‘₯ βˆ’ 𝑦)(,)(π‘₯ + 𝑦)) βŠ† 𝐴))
 
Theoremopnrebl2 35741* A set is open in the standard topology of the reals precisely when every point can be enclosed in an arbitrarily small ball. (Contributed by Jeff Hankins, 22-Sep-2013.) (Proof shortened by Mario Carneiro, 30-Jan-2014.)
(𝐴 ∈ (topGenβ€˜ran (,)) ↔ (𝐴 βŠ† ℝ ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ ℝ+ βˆƒπ‘§ ∈ ℝ+ (𝑧 ≀ 𝑦 ∧ ((π‘₯ βˆ’ 𝑧)(,)(π‘₯ + 𝑧)) βŠ† 𝐴)))
 
Theoremnn0prpwlem 35742* Lemma for nn0prpw 35743. Use strong induction to show that every positive integer has unique prime power divisors. (Contributed by Jeff Hankins, 28-Sep-2013.)
(𝐴 ∈ β„• β†’ βˆ€π‘˜ ∈ β„• (π‘˜ < 𝐴 β†’ βˆƒπ‘ ∈ β„™ βˆƒπ‘› ∈ β„• Β¬ ((𝑝↑𝑛) βˆ₯ π‘˜ ↔ (𝑝↑𝑛) βˆ₯ 𝐴)))
 
Theoremnn0prpw 35743* Two nonnegative integers are the same if and only if they are divisible by the same prime powers. (Contributed by Jeff Hankins, 29-Sep-2013.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ β„•0) β†’ (𝐴 = 𝐡 ↔ βˆ€π‘ ∈ β„™ βˆ€π‘› ∈ β„• ((𝑝↑𝑛) βˆ₯ 𝐴 ↔ (𝑝↑𝑛) βˆ₯ 𝐡)))
 
21.12.2  Basic topological facts
 
Theoremtopbnd 35744 Two equivalent expressions for the boundary of a topology. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) = (((clsβ€˜π½)β€˜π΄) βˆ– ((intβ€˜π½)β€˜π΄)))
 
Theoremopnbnd 35745 A set is open iff it is disjoint from its boundary. (Contributed by Jeff Hankins, 23-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ 𝐽 ↔ (𝐴 ∩ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴)))) = βˆ…))
 
Theoremcldbnd 35746 A set is closed iff it contains its boundary. (Contributed by Jeff Hankins, 1-Oct-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (𝐴 ∈ (Clsdβ€˜π½) ↔ (((clsβ€˜π½)β€˜π΄) ∩ ((clsβ€˜π½)β€˜(𝑋 βˆ– 𝐴))) βŠ† 𝐴))
 
Theoremntruni 35747* A union of interiors is a subset of the interior of the union. The reverse inclusion may not hold. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑂 βŠ† 𝒫 𝑋) β†’ βˆͺ π‘œ ∈ 𝑂 ((intβ€˜π½)β€˜π‘œ) βŠ† ((intβ€˜π½)β€˜βˆͺ 𝑂))
 
Theoremclsun 35748 A pairwise union of closures is the closure of the union. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ ((clsβ€˜π½)β€˜(𝐴 βˆͺ 𝐡)) = (((clsβ€˜π½)β€˜π΄) βˆͺ ((clsβ€˜π½)β€˜π΅)))
 
Theoremclsint2 35749* The closure of an intersection is a subset of the intersection of the closures. (Contributed by Jeff Hankins, 31-Aug-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐢 βŠ† 𝒫 𝑋) β†’ ((clsβ€˜π½)β€˜βˆ© 𝐢) βŠ† ∩ 𝑐 ∈ 𝐢 ((clsβ€˜π½)β€˜π‘))
 
Theoremopnregcld 35750* A set is regularly closed iff it is the closure of some open set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((clsβ€˜π½)β€˜((intβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘œ ∈ 𝐽 𝐴 = ((clsβ€˜π½)β€˜π‘œ)))
 
Theoremcldregopn 35751* A set if regularly open iff it is the interior of some closed set. (Contributed by Jeff Hankins, 27-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝐴 βŠ† 𝑋) β†’ (((intβ€˜π½)β€˜((clsβ€˜π½)β€˜π΄)) = 𝐴 ↔ βˆƒπ‘ ∈ (Clsdβ€˜π½)𝐴 = ((intβ€˜π½)β€˜π‘)))
 
Theoremneiin 35752 Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)
((𝐽 ∈ Top ∧ 𝑀 ∈ ((neiβ€˜π½)β€˜π΄) ∧ 𝑁 ∈ ((neiβ€˜π½)β€˜π΅)) β†’ (𝑀 ∩ 𝑁) ∈ ((neiβ€˜π½)β€˜(𝐴 ∩ 𝐡)))
 
Theoremhmeoclda 35753 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsdβ€˜π½)) β†’ (𝐹 β€œ 𝑆) ∈ (Clsdβ€˜πΎ))
 
Theoremhmeocldb 35754 Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.)
(((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽Homeo𝐾)) ∧ 𝑆 ∈ (Clsdβ€˜πΎ)) β†’ (◑𝐹 β€œ 𝑆) ∈ (Clsdβ€˜π½))
 
21.12.3  Topology of the real numbers
 
TheoremivthALT 35755* An alternate proof of the Intermediate Value Theorem ivth 25370 using topology. (Contributed by Jeff Hankins, 17-Aug-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ π‘ˆ ∈ ℝ) ∧ 𝐴 < 𝐡 ∧ ((𝐴[,]𝐡) βŠ† 𝐷 ∧ 𝐷 βŠ† β„‚ ∧ (𝐹 ∈ (𝐷–cnβ†’β„‚) ∧ (𝐹 β€œ (𝐴[,]𝐡)) βŠ† ℝ ∧ π‘ˆ ∈ ((πΉβ€˜π΄)(,)(πΉβ€˜π΅))))) β†’ βˆƒπ‘₯ ∈ (𝐴(,)𝐡)(πΉβ€˜π‘₯) = π‘ˆ)
 
21.12.4  Refinements
 
Syntaxcfne 35756 Extend class definition to include the "finer than" relation.
class Fne
 
Definitiondf-fne 35757* Define the fineness relation for covers. (Contributed by Jeff Hankins, 28-Sep-2009.)
Fne = {⟨π‘₯, π‘¦βŸ© ∣ (βˆͺ π‘₯ = βˆͺ 𝑦 ∧ βˆ€π‘§ ∈ π‘₯ 𝑧 βŠ† βˆͺ (𝑦 ∩ 𝒫 𝑧))}
 
Theoremfnerel 35758 Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Rel Fne
 
Theoremisfne 35759* The predicate "𝐡 is finer than 𝐴". This property is, in a sense, the opposite of refinement, as refinement requires every element to be a subset of an element of the original and fineness requires that every element of the original have a subset in the finer cover containing every point. I do not know of a literature reference for this. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 π‘₯ βŠ† βˆͺ (𝐡 ∩ 𝒫 π‘₯))))
 
Theoremisfne4 35760 The predicate "𝐡 is finer than 𝐴 " in terms of the topology generation function. (Contributed by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ 𝐴 βŠ† (topGenβ€˜π΅)))
 
Theoremisfne4b 35761 A condition for a topology to be finer than another. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝑉 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ (topGenβ€˜π΄) βŠ† (topGenβ€˜π΅))))
 
Theoremisfne2 35762* The predicate "𝐡 is finer than 𝐴". (Contributed by Jeff Hankins, 28-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘§ ∈ 𝐡 (𝑦 ∈ 𝑧 ∧ 𝑧 βŠ† π‘₯))))
 
Theoremisfne3 35763* The predicate "𝐡 is finer than 𝐴". (Contributed by Jeff Hankins, 11-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐡 ∈ 𝐢 β†’ (𝐴Fne𝐡 ↔ (𝑋 = π‘Œ ∧ βˆ€π‘₯ ∈ 𝐴 βˆƒπ‘¦(𝑦 βŠ† 𝐡 ∧ π‘₯ = βˆͺ 𝑦))))
 
Theoremfnebas 35764 A finer cover covers the same set as the original. (Contributed by Jeff Hankins, 28-Sep-2009.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝐴Fne𝐡 β†’ 𝑋 = π‘Œ)
 
Theoremfnetg 35765 A finer cover generates a topology finer than the original set. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝐴Fne𝐡 β†’ 𝐴 βŠ† (topGenβ€˜π΅))
 
Theoremfnessex 35766* If 𝐡 is finer than 𝐴 and 𝑆 is an element of 𝐴, every point in 𝑆 is an element of a subset of 𝑆 which is in 𝐡. (Contributed by Jeff Hankins, 28-Sep-2009.)
((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝑆) β†’ βˆƒπ‘₯ ∈ 𝐡 (𝑃 ∈ π‘₯ ∧ π‘₯ βŠ† 𝑆))
 
Theoremfneuni 35767* If 𝐡 is finer than 𝐴, every element of 𝐴 is a union of elements of 𝐡. (Contributed by Jeff Hankins, 11-Oct-2009.)
((𝐴Fne𝐡 ∧ 𝑆 ∈ 𝐴) β†’ βˆƒπ‘₯(π‘₯ βŠ† 𝐡 ∧ 𝑆 = βˆͺ π‘₯))
 
Theoremfneint 35768* If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)
(𝐴Fne𝐡 β†’ ∩ {π‘₯ ∈ 𝐡 ∣ 𝑃 ∈ π‘₯} βŠ† ∩ {π‘₯ ∈ 𝐴 ∣ 𝑃 ∈ π‘₯})
 
Theoremfness 35769 A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   ((𝐡 ∈ 𝐢 ∧ 𝐴 βŠ† 𝐡 ∧ 𝑋 = π‘Œ) β†’ 𝐴Fne𝐡)
 
Theoremfneref 35770 Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)
(𝐴 ∈ 𝑉 β†’ 𝐴Fne𝐴)
 
Theoremfnetr 35771 Transitivity of the fineness relation. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
((𝐴Fne𝐡 ∧ 𝐡Fne𝐢) β†’ 𝐴Fne𝐢)
 
Theoremfneval 35772 Two covers are finer than each other iff they are both bases for the same topology. (Contributed by Mario Carneiro, 11-Sep-2015.)
∼ = (Fne ∩ β—‘Fne)    β‡’   ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 ∼ 𝐡 ↔ (topGenβ€˜π΄) = (topGenβ€˜π΅)))
 
Theoremfneer 35773 Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
∼ = (Fne ∩ β—‘Fne)    β‡’    ∼ Er V
 
Theoremtopfne 35774 Fineness for covers corresponds precisely with fineness for topologies. (Contributed by Jeff Hankins, 29-Sep-2009.)
𝑋 = βˆͺ 𝐽    &   π‘Œ = βˆͺ 𝐾    β‡’   ((𝐾 ∈ Top ∧ 𝑋 = π‘Œ) β†’ (𝐽 βŠ† 𝐾 ↔ 𝐽Fne𝐾))
 
Theoremtopfneec 35775 A cover is equivalent to a topology iff it is a base for that topology. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
∼ = (Fne ∩ β—‘Fne)    β‡’   (𝐽 ∈ Top β†’ (𝐴 ∈ [𝐽] ∼ ↔ (topGenβ€˜π΄) = 𝐽))
 
Theoremtopfneec2 35776 A topology is precisely identified with its equivalence class. (Contributed by Jeff Hankins, 12-Oct-2009.)
∼ = (Fne ∩ β—‘Fne)    β‡’   ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) β†’ ([𝐽] ∼ = [𝐾] ∼ ↔ 𝐽 = 𝐾))
 
Theoremfnessref 35777* A cover is finer iff it has a subcover which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝑋 = π‘Œ β†’ (𝐴Fne𝐡 ↔ βˆƒπ‘(𝑐 βŠ† 𝐡 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
 
Theoremrefssfne 35778* A cover is a refinement iff it is a subcover of something which is both finer and a refinement. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)
𝑋 = βˆͺ 𝐴    &   π‘Œ = βˆͺ 𝐡    β‡’   (𝑋 = π‘Œ β†’ (𝐡Ref𝐴 ↔ βˆƒπ‘(𝐡 βŠ† 𝑐 ∧ (𝐴Fne𝑐 ∧ 𝑐Ref𝐴))))
 
21.12.5  Neighborhood bases determine topologies
 
Theoremneibastop1 35779* A collection of neighborhood bases determines a topology. Part of Theorem 4.5 of Stephen Willard's General Topology. (Contributed by Jeff Hankins, 8-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    β‡’   (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
 
Theoremneibastop2lem 35780* Lemma for neibastop2 35781. (Contributed by Jeff Hankins, 12-Sep-2009.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    &   (πœ‘ β†’ 𝑃 ∈ 𝑋)    &   (πœ‘ β†’ 𝑁 βŠ† 𝑋)    &   (πœ‘ β†’ π‘ˆ ∈ (πΉβ€˜π‘ƒ))    &   (πœ‘ β†’ π‘ˆ βŠ† 𝑁)    &   πΊ = (rec((π‘Ž ∈ V ↦ βˆͺ 𝑧 ∈ π‘Ž βˆͺ π‘₯ ∈ 𝑋 ((πΉβ€˜π‘₯) ∩ 𝒫 𝑧)), {π‘ˆ}) β†Ύ Ο‰)    &   π‘† = {𝑦 ∈ 𝑋 ∣ βˆƒπ‘“ ∈ βˆͺ ran 𝐺((πΉβ€˜π‘¦) ∩ 𝒫 𝑓) β‰  βˆ…}    β‡’   (πœ‘ β†’ βˆƒπ‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 ∧ 𝑒 βŠ† 𝑁))
 
Theoremneibastop2 35781* In the topology generated by a neighborhood base, a set is a neighborhood of a point iff it contains a subset in the base. (Contributed by Jeff Hankins, 9-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    β‡’   ((πœ‘ ∧ 𝑃 ∈ 𝑋) β†’ (𝑁 ∈ ((neiβ€˜π½)β€˜{𝑃}) ↔ (𝑁 βŠ† 𝑋 ∧ ((πΉβ€˜π‘ƒ) ∩ 𝒫 𝑁) β‰  βˆ…)))
 
Theoremneibastop3 35782* The topology generated by a neighborhood base is unique. (Contributed by Jeff Hankins, 16-Sep-2009.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)
(πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆ(𝒫 𝒫 𝑋 βˆ– {βˆ…}))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯) ∧ 𝑀 ∈ (πΉβ€˜π‘₯))) β†’ ((πΉβ€˜π‘₯) ∩ 𝒫 (𝑣 ∩ 𝑀)) β‰  βˆ…)    &   π½ = {π‘œ ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ π‘œ ((πΉβ€˜π‘₯) ∩ 𝒫 π‘œ) β‰  βˆ…}    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ π‘₯ ∈ 𝑣)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑣 ∈ (πΉβ€˜π‘₯))) β†’ βˆƒπ‘‘ ∈ (πΉβ€˜π‘₯)βˆ€π‘¦ ∈ 𝑑 ((πΉβ€˜π‘¦) ∩ 𝒫 𝑣) β‰  βˆ…)    β‡’   (πœ‘ β†’ βˆƒ!𝑗 ∈ (TopOnβ€˜π‘‹)βˆ€π‘₯ ∈ 𝑋 ((neiβ€˜π‘—)β€˜{π‘₯}) = {𝑛 ∈ 𝒫 𝑋 ∣ ((πΉβ€˜π‘₯) ∩ 𝒫 𝑛) β‰  βˆ…})
 
21.12.6  Lattice structure of topologies
 
Theoremtopmtcl 35783 The meet of a collection of topologies on 𝑋 is again a topology on 𝑋. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOnβ€˜π‘‹))
 
Theoremtopmeet 35784* Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (𝒫 𝑋 ∩ ∩ 𝑆) = βˆͺ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 π‘˜ βŠ† 𝑗})
 
Theoremtopjoin 35785* Two equivalent formulations of the join of a collection of topologies. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ 𝑆 βŠ† (TopOnβ€˜π‘‹)) β†’ (topGenβ€˜(fiβ€˜({𝑋} βˆͺ βˆͺ 𝑆))) = ∩ {π‘˜ ∈ (TopOnβ€˜π‘‹) ∣ βˆ€π‘— ∈ 𝑆 𝑗 βŠ† π‘˜})
 
Theoremfnemeet1 35786* The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦 ∧ 𝐴 ∈ 𝑆) β†’ (𝒫 𝑋 ∩ ∩ 𝑑 ∈ 𝑆 (topGenβ€˜π‘‘))Fne𝐴)
 
Theoremfnemeet2 35787* The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦) β†’ (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑑 ∈ 𝑆 (topGenβ€˜π‘‘)) ↔ (𝑋 = βˆͺ 𝑇 ∧ βˆ€π‘₯ ∈ 𝑆 𝑇Fneπ‘₯)))
 
Theoremfnejoin1 35788* Join of equivalence classes under the fineness relation-part one. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦 ∧ 𝐴 ∈ 𝑆) β†’ 𝐴Fneif(𝑆 = βˆ…, {𝑋}, βˆͺ 𝑆))
 
Theoremfnejoin2 35789* Join of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 8-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
((𝑋 ∈ 𝑉 ∧ βˆ€π‘¦ ∈ 𝑆 𝑋 = βˆͺ 𝑦) β†’ (if(𝑆 = βˆ…, {𝑋}, βˆͺ 𝑆)Fne𝑇 ↔ (𝑋 = βˆͺ 𝑇 ∧ βˆ€π‘₯ ∈ 𝑆 π‘₯Fne𝑇)))
 
21.12.7  Filter bases
 
Theoremfgmin 35790 Minimality property of a generated filter: every filter that contains 𝐡 contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
((𝐡 ∈ (fBasβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐡 βŠ† 𝐹 ↔ (𝑋filGen𝐡) βŠ† 𝐹))
 
Theoremneifg 35791* The neighborhood filter of a nonempty set is generated by its open supersets. See comments for opnfbas 23733. (Contributed by Jeff Hankins, 3-Sep-2009.)
𝑋 = βˆͺ 𝐽    β‡’   ((𝐽 ∈ Top ∧ 𝑆 βŠ† 𝑋 ∧ 𝑆 β‰  βˆ…) β†’ (𝑋filGen{π‘₯ ∈ 𝐽 ∣ 𝑆 βŠ† π‘₯}) = ((neiβ€˜π½)β€˜π‘†))
 
21.12.8  Directed sets, nets
 
Theoremtailfval 35792* The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   (𝐷 ∈ DirRel β†’ (tailβ€˜π·) = (π‘₯ ∈ 𝑋 ↦ (𝐷 β€œ {π‘₯})))
 
Theoremtailval 35793 The tail of an element in a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ ((tailβ€˜π·)β€˜π΄) = (𝐷 β€œ {𝐴}))
 
Theoremeltail 35794 An element of a tail. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝐢) β†’ (𝐡 ∈ ((tailβ€˜π·)β€˜π΄) ↔ 𝐴𝐷𝐡))
 
Theoremtailf 35795 The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)
𝑋 = dom 𝐷    β‡’   (𝐷 ∈ DirRel β†’ (tailβ€˜π·):π‘‹βŸΆπ’« 𝑋)
 
Theoremtailini 35796 A tail contains its initial element. (Contributed by Jeff Hankins, 25-Nov-2009.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ ((tailβ€˜π·)β€˜π΄))
 
Theoremtailfb 35797 The collection of tails of a directed set is a filter base. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝑋 = dom 𝐷    β‡’   ((𝐷 ∈ DirRel ∧ 𝑋 β‰  βˆ…) β†’ ran (tailβ€˜π·) ∈ (fBasβ€˜π‘‹))
 
Theoremfilnetlem1 35798* Lemma for filnet 35802. Change variables. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    &   π΄ ∈ V    &   π΅ ∈ V    β‡’   (𝐴𝐷𝐡 ↔ ((𝐴 ∈ 𝐻 ∧ 𝐡 ∈ 𝐻) ∧ (1st β€˜π΅) βŠ† (1st β€˜π΄)))
 
Theoremfilnetlem2 35799* Lemma for filnet 35802. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (( I β†Ύ 𝐻) βŠ† 𝐷 ∧ 𝐷 βŠ† (𝐻 Γ— 𝐻))
 
Theoremfilnetlem3 35800* Lemma for filnet 35802. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
𝐻 = βˆͺ 𝑛 ∈ 𝐹 ({𝑛} Γ— 𝑛)    &   π· = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st β€˜π‘¦) βŠ† (1st β€˜π‘₯))}    β‡’   (𝐻 = βˆͺ βˆͺ 𝐷 ∧ (𝐹 ∈ (Filβ€˜π‘‹) β†’ (𝐻 βŠ† (𝐹 Γ— 𝑋) ∧ 𝐷 ∈ DirRel)))
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48161
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