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Theorem List for Metamath Proof Explorer - 13901-14000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwwlktovf1 13901* Lemma 2 for wrd2f1tovbij 13904. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       𝐹:𝐷1-1𝑅

Theoremwwlktovfo 13902* Lemma 3 for wrd2f1tovbij 13904. (Contributed by Alexander van der Vekens, 27-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       (𝑃𝑉𝐹:𝐷onto𝑅)

Theoremwwlktovf1o 13903* Lemma 4 for wrd2f1tovbij 13904. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}    &   𝑅 = {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}    &   𝐹 = (𝑡𝐷 ↦ (𝑡‘1))       (𝑃𝑉𝐹:𝐷1-1-onto𝑅)

Theoremwrd2f1tovbij 13904* There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
((𝑉𝑌𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})

Theoremeqwrds3 13905 A word is equal with a length 3 string iff it has length 3 and the same symbol at each position. (Contributed by AV, 12-May-2021.)
((𝑊 ∈ Word 𝑉 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝑊 = ⟨“𝐴𝐵𝐶”⟩ ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = 𝐵 ∧ (𝑊‘2) = 𝐶))))

Theoremwrdl3s3 13906* A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Theorems3sndisj 13907* The singletons consisting of length 3 strings which have distinct third symbols are disjunct. (Contributed by AV, 17-May-2021.)
((𝐴𝑋𝐵𝑌) → Disj 𝑐𝑍 {⟨“𝐴𝐵𝑐”⟩})

Theorems3iunsndisj 13908* The union of singletons consisting of length 3 strings which have distinct first and third symbols are disjunct. (Contributed by AV, 17-May-2021.)
(𝐵𝑋Disj 𝑎𝑌 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Theoremofccat 13909 Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(𝜑𝐸 ∈ Word 𝑆)    &   (𝜑𝐹 ∈ Word 𝑆)    &   (𝜑𝐺 ∈ Word 𝑇)    &   (𝜑𝐻 ∈ Word 𝑇)    &   (𝜑 → (♯‘𝐸) = (♯‘𝐺))    &   (𝜑 → (♯‘𝐹) = (♯‘𝐻))       (𝜑 → ((𝐸 ++ 𝐹) ∘𝑓 𝑅(𝐺 ++ 𝐻)) = ((𝐸𝑓 𝑅𝐺) ++ (𝐹𝑓 𝑅𝐻)))

Theoremofs1 13910 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Theoremofs2 13911 Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
(((𝐴𝑆𝐵𝑆) ∧ (𝐶𝑇𝐷𝑇)) → (⟨“𝐴𝐵”⟩ ∘𝑓 𝑅⟨“𝐶𝐷”⟩) = ⟨“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”⟩)

5.8  Reflexive and transitive closures of relations

A relation, 𝑅, has the reflexive property if 𝐴𝑅𝐴 holds whenever 𝐴 is an element which could be related by the relation, namely elements of its domain and range. Eliminating dummy variables we see that a segment of the identity relation must be a subset of the relation or ( I ↾ (ran 𝑅 ∪ dom 𝑅)) ⊆ 𝑅. See issref 5667.

A relation, 𝑅, has the transitive property if 𝐴𝑅𝐶 holds whenever there exists an intermediate value 𝐵 such that both 𝐴𝑅𝐵 and 𝐵𝑅𝐶 hold. This can be expressed without dummy variables as (𝑅𝑅) ⊆ 𝑅. See cotr 5666.

The transitive closure of a relation, (t+‘𝑅), is the smallest superset of the relation which has the transitive property. Likewise the reflexive-transitive closure, (t*‘𝑅), is the smallest superset which has both the reflexive and transitive properties.

Not to be confused with the transitive closure of a set, trcl 8777, which is a closure relative to a different transitive property, df-tr 4905.

5.8.1  The reflexive and transitive properties of relations

Theoremcoss12d 13912 Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremtrrelssd 13913 The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆𝑅)    &   (𝜑𝑇𝑅)       (𝜑 → (𝑆𝑇) ⊆ 𝑅)

Theoremxpcogend 13914 The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝐵𝐶) ≠ ∅)       (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Theoremxpcoidgend 13915 If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝐴𝐵) ≠ ∅)       (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵))

Theoremcotr2g 13916* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 13917 for the main application. (Contributed by RP, 22-Mar-2020.)
dom 𝐵𝐷    &   (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸    &   ran 𝐴𝐹       ((𝐴𝐵) ⊆ 𝐶 ↔ ∀𝑥𝐷𝑦𝐸𝑧𝐹 ((𝑥𝐵𝑦𝑦𝐴𝑧) → 𝑥𝐶𝑧))

Theoremcotr2 13917* Two ways of saying a relation is transitive. Special instance of cotr2g 13916. (Contributed by RP, 22-Mar-2020.)
dom 𝑅𝐴    &   (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵    &   ran 𝑅𝐶       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremcotr3 13918* Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
𝐴 = dom 𝑅    &   𝐵 = (𝐴𝐶)    &   𝐶 = ran 𝑅       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremcoemptyd 13919 Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremxptrrel 13920 The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)

Theorem0trrel 13921 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(∅ ∘ ∅) ⊆ ∅

5.8.2  Basic properties of closures

Theoremcleq1lem 13922 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))

Theoremcleq1 13923* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})

Theoremclsslem 13924* The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} ⊆ {𝑟 ∣ (𝑆𝑟𝜑)})

5.8.3  Definitions and basic properties of transitive closures

Syntaxctcl 13925 Extend class notation to include the transitive closure symbol.
class t+

Syntaxcrtcl 13926 Extend class notation with reflexive-transitive closure.
class t*

Definitiondf-trcl 13927* Transitive closure of a relation. This is the smallest superset which has the transitive property. (Contributed by FL, 27-Jun-2011.)
t+ = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Definitiondf-rtrcl 13928* 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* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})

Theoremtrcleq1 13929* Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} = {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrclsslem 13930* The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
(𝑅𝑆 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ {𝑟 ∣ (𝑆𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})

Theoremtrcleq2lem 13931 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremcvbtrcl 13932* Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
{𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}

Theoremtrcleq12lem 13933 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremtrclexlem 13934 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V)

Theoremtrclublem 13935* 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.)
(𝑅𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theoremtrclubi 13936* 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    &   𝑅 ∈ V        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)

Theoremtrclubgi 13937* 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        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))

Theoremtrclub 13938* 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.)
((𝑅𝑉 ∧ Rel 𝑅) → {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (dom 𝑅 × ran 𝑅))

Theoremtrclubg 13939* 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.)
(𝑅𝑉 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfv 13940* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theorembrintclab 13941* Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)
(𝐴 {𝑥𝜑}𝐵 ↔ ∀𝑥(𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑥))

Theorembrtrclfv 13942* Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfv 13943* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theorembrtrclfvcnv 13944* Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
(𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))

Theorembrcnvtrclfvcnv 13945* Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.)
((𝑅𝑈𝐴𝑉𝐵𝑊) → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴)))

Theoremtrclfvss 13946 The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.)
((𝑅𝑉𝑆𝑊𝑅𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆))

Theoremtrclfvub 13947 The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

Theoremtrclfvlb 13948 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcotr 13949 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Theoremtrclfvlb2 13950 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))

Theoremtrclfvlb3 13951 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))

Theoremcotrtrclfv 13952 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Theoremtrclidm 13953 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))

Theoremtrclun 13954 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Theoremtrclfvg 13955 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 ⊆ (t+‘𝑅) ∨ (t+‘𝑅) = ∅)

Theoremtrclfvcotrg 13956 The value of the transitive closure of a relation is always a transitive relation. (Contributed by RP, 8-May-2020.)
((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)

Theoremreltrclfv 13957 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Theoremdmtrclfv 13958 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.)
(𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)

5.8.4  Exponentiation of relations

Syntaxcrelexp 13959 Extend class notation to include relation exponentiation.
class 𝑟

Definitiondf-relexp 13960* Definition of repeated composition of a relation with itself, aka relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 22-May-2020.)
𝑟 = (𝑟 ∈ V, 𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( I ↾ (dom 𝑟 ∪ ran 𝑟)), (seq1((𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑥𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛)))

Theoremrelexp0g 13961 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))

Theoremrelexp0 13962 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexp0d 13963 A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexpsucnnr 13964 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexp1g 13965 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟1) = 𝑅)

Theoremdfid5 13966 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ (𝑥𝑟1))

Theoremdfid6 13967* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ 𝑛 ∈ {1} (𝑥𝑟𝑛))

Theoremrelexpsucr 13968 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexpsucrd 13969 A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅)))

Theoremrelexp1d 13970 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟1) = 𝑅)

Theoremrelexpsucnnl 13971 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucl 13972 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucld 13973 A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁))))

Theoremrelexpcnv 13974 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))

Theoremrelexpcnvd 13975 Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0(𝑅𝑟𝑁) = (𝑅𝑟𝑁)))

Theoremrelexp0rel 13976 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅𝑉 → Rel (𝑅𝑟0))

Theoremrelexprelg 13977 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 ∈ ℕ0𝑅𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅𝑟𝑁))

Theoremrelexprel 13978 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉 ∧ Rel 𝑅) → Rel (𝑅𝑟𝑁))

Theoremrelexpreld 13979 The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → Rel (𝑅𝑟𝑁)))

Theoremrelexpnndm 13980 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ dom 𝑅)

Theoremrelexpdmg 13981 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))

Theoremrelexpdm 13982 The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → dom (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexpdmd 13983 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → dom (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpnnrn 13984 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ ran 𝑅)

Theoremrelexprng 13985 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))

Theoremrelexprn 13986 The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → ran (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexprnd 13987 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 → ran (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpfld 13988 The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) ⊆ 𝑅)

Theoremrelexpfldd 13989 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑁 ∈ ℕ0 (𝑅𝑟𝑁) ⊆ 𝑅))

Theoremrelexpaddnn 13990 Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Theoremrelexpuzrel 13991 The exponentiation of a class to an integer not smaller than 2 is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑅𝑉) → Rel (𝑅𝑟𝑁))

Theoremrelexpaddg 13992 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 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑅𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀)))

Theoremrelexpaddd 13993 Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ((𝑁 ∈ ℕ0𝑀 ∈ ℕ0) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) = (𝑅𝑟(𝑁 + 𝑀))))

5.8.5  Reflexive-transitive closure as an indexed union

Syntaxcrtrcl 13994 Extend class notation with recursively defined reflexive, transitive closure.
class t*rec

Definitiondf-rtrclrec 13995* The reflexive, transitive closure of a relation constructed as the union of all finite exponentiations. (Contributed by Drahflow, 12-Nov-2015.)
t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))

Theoremdfrtrclrec2 13996* 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.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))

Theoremrtrclreclem1 13997 The reflexive, transitive closure is indeed reflexive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (t*rec‘𝑅))

Theoremrtrclreclem2 13998 The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t*rec‘𝑅))

Theoremrtrclreclem3 13999 The reflexive, transitive closure is indeed transitive. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))

Theoremrtrclreclem4 14000* 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.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))

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