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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
5.8.2  Basic properties of closures
 
Theoremcleq1lem 14901 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐡 β†’ ((𝐴 βŠ† 𝐢 ∧ πœ‘) ↔ (𝐡 βŠ† 𝐢 ∧ πœ‘)))
 
Theoremcleq1 14902* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ πœ‘)} = ∩ {π‘Ÿ ∣ (𝑆 βŠ† π‘Ÿ ∧ πœ‘)})
 
Theoremclsslem 14903* The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
(𝑅 βŠ† 𝑆 β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ πœ‘)} βŠ† ∩ {π‘Ÿ ∣ (𝑆 βŠ† π‘Ÿ ∧ πœ‘)})
 
5.8.3  Definitions and basic properties of transitive closures
 
Syntaxctcl 14904 Extend class notation to include the transitive closure symbol.
class t+
 
Syntaxcrtcl 14905 Extend class notation with reflexive-transitive closure.
class t*
 
Definitiondf-trcl 14906* Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
t+ = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
 
Definitiondf-rtrcl 14907* Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.)
t* = (π‘₯ ∈ V ↦ ∩ {𝑧 ∣ (( I β†Ύ (dom π‘₯ βˆͺ ran π‘₯)) βŠ† 𝑧 ∧ π‘₯ βŠ† 𝑧 ∧ (𝑧 ∘ 𝑧) βŠ† 𝑧)})
 
Theoremtrcleq1 14908* Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)} = ∩ {π‘Ÿ ∣ (𝑆 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)})
 
Theoremtrclsslem 14909* The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
(𝑅 βŠ† 𝑆 β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)} βŠ† ∩ {π‘Ÿ ∣ (𝑆 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)})
 
Theoremtrcleq2lem 14910 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐡 β†’ ((𝑅 βŠ† 𝐴 ∧ (𝐴 ∘ 𝐴) βŠ† 𝐴) ↔ (𝑅 βŠ† 𝐡 ∧ (𝐡 ∘ 𝐡) βŠ† 𝐡)))
 
Theoremcvbtrcl 14911* Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
{π‘₯ ∣ (𝑅 βŠ† π‘₯ ∧ (π‘₯ ∘ π‘₯) βŠ† π‘₯)} = {𝑦 ∣ (𝑅 βŠ† 𝑦 ∧ (𝑦 ∘ 𝑦) βŠ† 𝑦)}
 
Theoremtrcleq12lem 14912 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆 ∧ 𝐴 = 𝐡) β†’ ((𝑅 βŠ† 𝐴 ∧ (𝐴 ∘ 𝐴) βŠ† 𝐴) ↔ (𝑆 βŠ† 𝐡 ∧ (𝐡 ∘ 𝐡) βŠ† 𝐡)))
 
Theoremtrclexlem 14913 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
(𝑅 ∈ 𝑉 β†’ (𝑅 βˆͺ (dom 𝑅 Γ— ran 𝑅)) ∈ V)
 
Theoremtrclublem 14914* If a relation exists then the class of transitive relations which are supersets of that relation is not empty. (Contributed by RP, 28-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ (𝑅 βˆͺ (dom 𝑅 Γ— ran 𝑅)) ∈ {π‘₯ ∣ (𝑅 βŠ† π‘₯ ∧ (π‘₯ ∘ π‘₯) βŠ† π‘₯)})
 
Theoremtrclubi 14915* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 2-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
Rel 𝑅    &   π‘… ∈ V    β‡’   βˆ© {𝑠 ∣ (𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠)} βŠ† (dom 𝑅 Γ— ran 𝑅)
 
Theoremtrclubgi 14916* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure. (Contributed by RP, 3-Jan-2020.) (Revised by RP, 28-Apr-2020.) (Revised by AV, 26-Mar-2021.)
𝑅 ∈ V    β‡’   βˆ© {𝑠 ∣ (𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠)} βŠ† (𝑅 βˆͺ (dom 𝑅 Γ— ran 𝑅))
 
Theoremtrclub 14917* The Cartesian product of the domain and range of a relation is an upper bound for its transitive closure. (Contributed by RP, 17-May-2020.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅) β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)} βŠ† (dom 𝑅 Γ— ran 𝑅))
 
Theoremtrclubg 14918* The union with the Cartesian product of its domain and range is an upper bound for a set's transitive closure (as a relation). (Contributed by RP, 17-May-2020.)
(𝑅 ∈ 𝑉 β†’ ∩ {π‘Ÿ ∣ (𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ)} βŠ† (𝑅 βˆͺ (dom 𝑅 Γ— ran 𝑅)))
 
Theoremtrclfv 14919* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ (t+β€˜π‘…) = ∩ {π‘₯ ∣ (𝑅 βŠ† π‘₯ ∧ (π‘₯ ∘ π‘₯) βŠ† π‘₯)})
 
Theorembrintclab 14920* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
(𝐴∩ {π‘₯ ∣ πœ‘}𝐡 ↔ βˆ€π‘₯(πœ‘ β†’ ⟨𝐴, 𝐡⟩ ∈ π‘₯))
 
Theorembrtrclfv 14921* Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
(𝑅 ∈ 𝑉 β†’ (𝐴(t+β€˜π‘…)𝐡 ↔ βˆ€π‘Ÿ((𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ) β†’ π΄π‘Ÿπ΅)))
 
Theorembrcnvtrclfv 14922* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.)
((𝑅 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴◑(t+β€˜π‘…)𝐡 ↔ βˆ€π‘Ÿ((𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ) β†’ π΅π‘Ÿπ΄)))
 
Theorembrtrclfvcnv 14923* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
(𝑅 ∈ 𝑉 β†’ (𝐴(t+β€˜β—‘π‘…)𝐡 ↔ βˆ€π‘Ÿ((◑𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ) β†’ π΄π‘Ÿπ΅)))
 
Theorembrcnvtrclfvcnv 14924* Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
((𝑅 ∈ π‘ˆ ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴◑(t+β€˜β—‘π‘…)𝐡 ↔ βˆ€π‘Ÿ((◑𝑅 βŠ† π‘Ÿ ∧ (π‘Ÿ ∘ π‘Ÿ) βŠ† π‘Ÿ) β†’ π΅π‘Ÿπ΄)))
 
Theoremtrclfvss 14925 The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š ∧ 𝑅 βŠ† 𝑆) β†’ (t+β€˜π‘…) βŠ† (t+β€˜π‘†))
 
Theoremtrclfvub 14926 The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ (t+β€˜π‘…) βŠ† (𝑅 βˆͺ (dom 𝑅 Γ— ran 𝑅)))
 
Theoremtrclfvlb 14927 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ 𝑅 βŠ† (t+β€˜π‘…))
 
Theoremtrclfvcotr 14928 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ ((t+β€˜π‘…) ∘ (t+β€˜π‘…)) βŠ† (t+β€˜π‘…))
 
Theoremtrclfvlb2 14929 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅 ∈ 𝑉 β†’ (𝑅 ∘ 𝑅) βŠ† (t+β€˜π‘…))
 
Theoremtrclfvlb3 14930 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅 ∈ 𝑉 β†’ (𝑅 βˆͺ (𝑅 ∘ 𝑅)) βŠ† (t+β€˜π‘…))
 
Theoremcotrtrclfv 14931 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) βŠ† 𝑅) β†’ (t+β€˜π‘…) = 𝑅)
 
Theoremtrclidm 14932 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅 ∈ 𝑉 β†’ (t+β€˜(t+β€˜π‘…)) = (t+β€˜π‘…))
 
Theoremtrclun 14933 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ π‘Š) β†’ (t+β€˜(𝑅 βˆͺ 𝑆)) = (t+β€˜((t+β€˜π‘…) βˆͺ (t+β€˜π‘†))))
 
Theoremtrclfvg 14934 The value of the transitive closure of a relation is a superset or (for proper classes) the empty set. (Contributed by RP, 8-May-2020.)
(𝑅 βŠ† (t+β€˜π‘…) ∨ (t+β€˜π‘…) = βˆ…)
 
Theoremtrclfvcotrg 14935 The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.)
((t+β€˜π‘…) ∘ (t+β€˜π‘…)) βŠ† (t+β€˜π‘…)
 
Theoremreltrclfv 14936 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅) β†’ Rel (t+β€˜π‘…))
 
Theoremdmtrclfv 14937 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)
(𝑅 ∈ 𝑉 β†’ dom (t+β€˜π‘…) = dom 𝑅)
 
5.8.4  Exponentiation of relations
 
Syntaxcrelexp 14938 Extend class notation to include relation exponentiation.
class β†‘π‘Ÿ
 
Definitiondf-relexp 14939* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.)
β†‘π‘Ÿ = (π‘Ÿ ∈ V, 𝑛 ∈ β„•0 ↦ if(𝑛 = 0, ( I β†Ύ (dom π‘Ÿ βˆͺ ran π‘Ÿ)), (seq1((π‘₯ ∈ V, 𝑦 ∈ V ↦ (π‘₯ ∘ π‘Ÿ)), (𝑧 ∈ V ↦ π‘Ÿ))β€˜π‘›)))
 
Theoremreldmrelexp 14940 The domain of the repeated composition of a relation is a relation. (Contributed by AV, 12-Jul-2024.)
Rel dom β†‘π‘Ÿ
 
Theoremrelexp0g 14941 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅 ∈ 𝑉 β†’ (π‘…β†‘π‘Ÿ0) = ( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)))
 
Theoremrelexp0 14942 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅) β†’ (π‘…β†‘π‘Ÿ0) = ( I β†Ύ βˆͺ βˆͺ 𝑅))
 
Theoremrelexp0d 14943 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘…β†‘π‘Ÿ0) = ( I β†Ύ βˆͺ βˆͺ 𝑅))
 
Theoremrelexpsucnnr 14944 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = ((π‘…β†‘π‘Ÿπ‘) ∘ 𝑅))
 
Theoremrelexp1g 14945 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅 ∈ 𝑉 β†’ (π‘…β†‘π‘Ÿ1) = 𝑅)
 
Theoremdfid5 14946 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (π‘₯ ∈ V ↦ (π‘₯β†‘π‘Ÿ1))
 
Theoremdfid6 14947* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (π‘₯ ∈ V ↦ βˆͺ 𝑛 ∈ {1} (π‘₯β†‘π‘Ÿπ‘›))
 
Theoremrelexp1d 14948 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑅 ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘…β†‘π‘Ÿ1) = 𝑅)
 
Theoremrelexpsucnnl 14949 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = (𝑅 ∘ (π‘…β†‘π‘Ÿπ‘)))
 
Theoremrelexpsucl 14950 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ β„•0) β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = (𝑅 ∘ (π‘…β†‘π‘Ÿπ‘)))
 
Theoremrelexpsucr 14951 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ β„•0) β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = ((π‘…β†‘π‘Ÿπ‘) ∘ 𝑅))
 
Theoremrelexpsucrd 14952 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = ((π‘…β†‘π‘Ÿπ‘) ∘ 𝑅))
 
Theoremrelexpsucld 14953 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ (π‘…β†‘π‘Ÿ(𝑁 + 1)) = (𝑅 ∘ (π‘…β†‘π‘Ÿπ‘)))
 
Theoremrelexpcnv 14954 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ β—‘(π‘…β†‘π‘Ÿπ‘) = (β—‘π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexpcnvd 14955 Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑅 ∈ 𝑉)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ β—‘(π‘…β†‘π‘Ÿπ‘) = (β—‘π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexp0rel 14956 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅 ∈ 𝑉 β†’ Rel (π‘…β†‘π‘Ÿ0))
 
Theoremrelexprelg 14957 The exponentiation of a class is a relation except when the exponent is one and the class is not a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 β†’ Rel 𝑅)) β†’ Rel (π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexprel 14958 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉 ∧ Rel 𝑅) β†’ Rel (π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexpreld 14959 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ Rel (π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexpnndm 14960 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„• ∧ 𝑅 ∈ 𝑉) β†’ dom (π‘…β†‘π‘Ÿπ‘) βŠ† dom 𝑅)
 
Theoremrelexpdmg 14961 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ dom (π‘…β†‘π‘Ÿπ‘) βŠ† (dom 𝑅 βˆͺ ran 𝑅))
 
Theoremrelexpdm 14962 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ dom (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexpdmd 14963 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ dom (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexpnnrn 14964 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„• ∧ 𝑅 ∈ 𝑉) β†’ ran (π‘…β†‘π‘Ÿπ‘) βŠ† ran 𝑅)
 
Theoremrelexprng 14965 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ ran (π‘…β†‘π‘Ÿπ‘) βŠ† (dom 𝑅 βˆͺ ran 𝑅))
 
Theoremrelexprn 14966 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ ran (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexprnd 14967 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ ran (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexpfld 14968 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„•0 ∧ 𝑅 ∈ 𝑉) β†’ βˆͺ βˆͺ (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexpfldd 14969 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    β‡’   (πœ‘ β†’ βˆͺ βˆͺ (π‘…β†‘π‘Ÿπ‘) βŠ† βˆͺ βˆͺ 𝑅)
 
Theoremrelexpaddnn 14970 Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ β„• ∧ 𝑀 ∈ β„• ∧ 𝑅 ∈ 𝑉) β†’ ((π‘…β†‘π‘Ÿπ‘) ∘ (π‘…β†‘π‘Ÿπ‘€)) = (π‘…β†‘π‘Ÿ(𝑁 + 𝑀)))
 
Theoremrelexpuzrel 14971 The exponentiation of a class to an integer not smaller than 2 is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ (β„€β‰₯β€˜2) ∧ 𝑅 ∈ 𝑉) β†’ Rel (π‘…β†‘π‘Ÿπ‘))
 
Theoremrelexpaddg 14972 Relation composition becomes addition under exponentiation except when the exponents total to one and the class isn't a relation. (Contributed by RP, 30-May-2020.)
((𝑁 ∈ β„•0 ∧ (𝑀 ∈ β„•0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 β†’ Rel 𝑅))) β†’ ((π‘…β†‘π‘Ÿπ‘) ∘ (π‘…β†‘π‘Ÿπ‘€)) = (π‘…β†‘π‘Ÿ(𝑁 + 𝑀)))
 
Theoremrelexpaddd 14973 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝑀 ∈ β„•0)    β‡’   (πœ‘ β†’ ((π‘…β†‘π‘Ÿπ‘) ∘ (π‘…β†‘π‘Ÿπ‘€)) = (π‘…β†‘π‘Ÿ(𝑁 + 𝑀)))
 
5.8.5  Reflexive-transitive closure as an indexed union
 
Syntaxcrtrcl 14974 Extend class notation with recursively defined reflexive, transitive closure.
class t*rec
 
Definitiondf-rtrclrec 14975* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))
 
Theoremrtrclreclem1 14976 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
(πœ‘ β†’ 𝑅 ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑅 βŠ† (t*recβ€˜π‘…))
 
Theoremdfrtrclrec2 14977* If two elements are connected by a reflexive, transitive closure, then they are connected via 𝑛 instances the relation, for some 𝑛. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    β‡’   (πœ‘ β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
 
Theoremrtrclreclem2 14978 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    &   (πœ‘ β†’ 𝑅 ∈ 𝑉)    β‡’   (πœ‘ β†’ ( I β†Ύ βˆͺ βˆͺ 𝑅) βŠ† (t*recβ€˜π‘…))
 
Theoremrtrclreclem3 14979 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    β‡’   (πœ‘ β†’ ((t*recβ€˜π‘…) ∘ (t*recβ€˜π‘…)) βŠ† (t*recβ€˜π‘…))
 
Theoremrtrclreclem4 14980* The reflexive, transitive closure of 𝑅 is the smallest reflexive, transitive relation which contains 𝑅 and the identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    β‡’   (πœ‘ β†’ βˆ€π‘ ((( I β†Ύ (dom 𝑅 βˆͺ ran 𝑅)) βŠ† 𝑠 ∧ 𝑅 βŠ† 𝑠 ∧ (𝑠 ∘ 𝑠) βŠ† 𝑠) β†’ (t*recβ€˜π‘…) βŠ† 𝑠))
 
Theoremdfrtrcl2 14981 The two definitions t* and t*rec of the reflexive, transitive closure coincide if 𝑅 is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(πœ‘ β†’ Rel 𝑅)    β‡’   (πœ‘ β†’ (t*β€˜π‘…) = (t*recβ€˜π‘…))
 
5.8.6  Principle of transitive induction.

If we have a statement that holds for some element, and a relation between elements that implies if it holds for the first element then it must hold for the second element, the principle of transitive induction shows the statement holds for any element related to the first by the (reflexive-)transitive closure of the relation.

 
Theoremrelexpindlem 14982* Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (Revised by AV, 13-Jul-2024.)
(πœ‚ β†’ Rel 𝑅)    &   (πœ‚ β†’ 𝑆 ∈ 𝑉)    &   (𝑖 = 𝑆 β†’ (πœ‘ ↔ πœ’))    &   (𝑖 = π‘₯ β†’ (πœ‘ ↔ πœ“))    &   (𝑖 = 𝑗 β†’ (πœ‘ ↔ πœƒ))    &   (πœ‚ β†’ πœ’)    &   (πœ‚ β†’ (𝑗𝑅π‘₯ β†’ (πœƒ β†’ πœ“)))    β‡’   (πœ‚ β†’ (𝑛 ∈ β„•0 β†’ (𝑆(π‘…β†‘π‘Ÿπ‘›)π‘₯ β†’ πœ“)))
 
Theoremrelexpind 14983* Principle of transitive induction, finite version. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
(πœ‚ β†’ Rel 𝑅)    &   (πœ‚ β†’ 𝑆 ∈ 𝑉)    &   (πœ‚ β†’ 𝑋 ∈ π‘Š)    &   (𝑖 = 𝑆 β†’ (πœ‘ ↔ πœ’))    &   (𝑖 = π‘₯ β†’ (πœ‘ ↔ πœ“))    &   (𝑖 = 𝑗 β†’ (πœ‘ ↔ πœƒ))    &   (π‘₯ = 𝑋 β†’ (πœ“ ↔ 𝜏))    &   (πœ‚ β†’ πœ’)    &   (πœ‚ β†’ (𝑗𝑅π‘₯ β†’ (πœƒ β†’ πœ“)))    β‡’   (πœ‚ β†’ (𝑛 ∈ β„•0 β†’ (𝑆(π‘…β†‘π‘Ÿπ‘›)𝑋 β†’ 𝜏)))
 
Theoremrtrclind 14984* Principle of transitive induction. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)
(πœ‚ β†’ Rel 𝑅)    &   (πœ‚ β†’ 𝑆 ∈ 𝑉)    &   (πœ‚ β†’ 𝑋 ∈ π‘Š)    &   (𝑖 = 𝑆 β†’ (πœ‘ ↔ πœ’))    &   (𝑖 = π‘₯ β†’ (πœ‘ ↔ πœ“))    &   (𝑖 = 𝑗 β†’ (πœ‘ ↔ πœƒ))    &   (π‘₯ = 𝑋 β†’ (πœ“ ↔ 𝜏))    &   (πœ‚ β†’ πœ’)    &   (πœ‚ β†’ (𝑗𝑅π‘₯ β†’ (πœƒ β†’ πœ“)))    β‡’   (πœ‚ β†’ (𝑆(t*β€˜π‘…)𝑋 β†’ 𝜏))
 
5.9  Elementary real and complex functions
 
5.9.1  The "shift" operation
 
Syntaxcshi 14985 Extend class notation with function shifter.
class shift
 
Definitiondf-shft 14986* Define a function shifter. This operation offsets the value argument of a function (ordinarily on a subset of β„‚) and produces a new function on β„‚. See shftval 14993 for its value. (Contributed by NM, 20-Jul-2005.)
shift = (𝑓 ∈ V, π‘₯ ∈ β„‚ ↦ {βŸ¨π‘¦, π‘§βŸ© ∣ (𝑦 ∈ β„‚ ∧ (𝑦 βˆ’ π‘₯)𝑓𝑧)})
 
Theoremshftlem 14987* Two ways to write a shifted set (𝐡 + 𝐴). (Contributed by Mario Carneiro, 3-Nov-2013.)
((𝐴 ∈ β„‚ ∧ 𝐡 βŠ† β„‚) β†’ {π‘₯ ∈ β„‚ ∣ (π‘₯ βˆ’ 𝐴) ∈ 𝐡} = {π‘₯ ∣ βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (𝑦 + 𝐴)})
 
Theoremshftuz 14988* A shift of the upper integers. (Contributed by Mario Carneiro, 5-Nov-2013.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€) β†’ {π‘₯ ∈ β„‚ ∣ (π‘₯ βˆ’ 𝐴) ∈ (β„€β‰₯β€˜π΅)} = (β„€β‰₯β€˜(𝐡 + 𝐴)))
 
Theoremshftfval 14989* The value of the sequence shifter operation is a function on β„‚. 𝐴 is ordinarily an integer. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V    β‡’   (𝐴 ∈ β„‚ β†’ (𝐹 shift 𝐴) = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ β„‚ ∧ (π‘₯ βˆ’ 𝐴)𝐹𝑦)})
 
Theoremshftdm 14990* Domain of a relation shifted by 𝐴. The set on the right is more commonly notated as (dom 𝐹 + 𝐴) (meaning add 𝐴 to every element of dom 𝐹). (Contributed by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V    β‡’   (𝐴 ∈ β„‚ β†’ dom (𝐹 shift 𝐴) = {π‘₯ ∈ β„‚ ∣ (π‘₯ βˆ’ 𝐴) ∈ dom 𝐹})
 
Theoremshftfib 14991 Value of a fiber of the relation 𝐹. (Contributed by Mario Carneiro, 4-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift 𝐴) β€œ {𝐡}) = (𝐹 β€œ {(𝐡 βˆ’ 𝐴)}))
 
Theoremshftfn 14992* Functionality and domain of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐹 Fn 𝐡 ∧ 𝐴 ∈ β„‚) β†’ (𝐹 shift 𝐴) Fn {π‘₯ ∈ β„‚ ∣ (π‘₯ βˆ’ 𝐴) ∈ 𝐡})
 
Theoremshftval 14993 Value of a sequence shifted by 𝐴. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 4-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift 𝐴)β€˜π΅) = (πΉβ€˜(𝐡 βˆ’ 𝐴)))
 
Theoremshftval2 14994 Value of a sequence shifted by 𝐴 βˆ’ 𝐡. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ β„‚) β†’ ((𝐹 shift (𝐴 βˆ’ 𝐡))β€˜(𝐴 + 𝐢)) = (πΉβ€˜(𝐡 + 𝐢)))
 
Theoremshftval3 14995 Value of a sequence shifted by 𝐴 βˆ’ 𝐡. (Contributed by NM, 20-Jul-2005.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift (𝐴 βˆ’ 𝐡))β€˜π΄) = (πΉβ€˜π΅))
 
Theoremshftval4 14996 Value of a sequence shifted by -𝐴. (Contributed by NM, 18-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift -𝐴)β€˜π΅) = (πΉβ€˜(𝐴 + 𝐡)))
 
Theoremshftval5 14997 Value of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift 𝐴)β€˜(𝐡 + 𝐴)) = (πΉβ€˜π΅))
 
Theoremshftf 14998* Functionality of a shifted sequence. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐹:𝐡⟢𝐢 ∧ 𝐴 ∈ β„‚) β†’ (𝐹 shift 𝐴):{π‘₯ ∈ β„‚ ∣ (π‘₯ βˆ’ 𝐴) ∈ 𝐡}⟢𝐢)
 
Theorem2shfti 14999 Composite shift operations. (Contributed by NM, 19-Aug-2005.) (Revised by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐹 shift 𝐴) shift 𝐡) = (𝐹 shift (𝐴 + 𝐡)))
 
Theoremshftidt2 15000 Identity law for the shift operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
𝐹 ∈ V    β‡’   (𝐹 shift 0) = (𝐹 β†Ύ β„‚)
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