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Theorem List for Metamath Proof Explorer - 13401-13500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwrd3tpop 13401 A word of length 3 represented as triple of ordered pairs. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩, ⟨2, (𝑊‘2)⟩})
 
Theoremwrdlen3s3 13402 A word of length 3 as length 3 string. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 3) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)(𝑊‘2)”⟩)
 
Theoremrepsw2 13403 The "repeated symbol word" of length 2. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 2) = ⟨“𝑆𝑆”⟩)
 
Theoremrepsw3 13404 The "repeated symbol word" of length 3. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 3) = ⟨“𝑆𝑆𝑆”⟩)
 
Theoremswrd2lsw 13405 Extract the last two symbols from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 substr ⟨((#‘𝑊) − 2), (#‘𝑊)⟩) = ⟨“(𝑊‘((#‘𝑊) − 2))( lastS ‘𝑊)”⟩)
 
Theorem2swrd2eqwrdeq 13406 Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 1 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 2)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 2)⟩) ∧ (𝑊‘((#‘𝑊) − 2)) = (𝑈‘((#‘𝑊) − 2)) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
 
Theoremccatw2s1ccatws2 13407 The concatenation of a word with two singleton words equals the concatenation of the word with the doubleton word consisting of the symbols of the two singletons. (Contributed by Mario Carneiro/AV, 21-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))
 
Theoremccat2s1fvwALT 13408 Alternate proof of ccat2s1fvw 13129 using words of length 2, see df-s2 13306. A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018.) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (#‘𝑊)) ∧ (𝑋𝑉𝑌𝑉)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))
 
Theoremwwlktovf 13409* Lemma 1 for wrd2f1tovbij 13413. (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:𝐷𝑅
 
Theoremwwlktovf1 13410* Lemma 2 for wrd2f1tovbij 13413. (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 13411* Lemma 3 for wrd2f1tovbij 13413. (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 13412* Lemma 4 for wrd2f1tovbij 13413. (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 13413* 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 13414 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 13415* 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 13416* 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 13417* 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 13418 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 13419 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘𝑓 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)
 
Theoremofs2 13420 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 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 5319.

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 5318.

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 8367, which is a closure relative to a different transitive property, df-tr 4579.

 
5.8.1  The reflexive and transitive properties of relations
 
Theoremcoss12d 13421 Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))
 
Theoremtrrelssd 13422 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 13423 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 13424 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 13425* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 13426 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 13426* Two ways of saying a relation is transitive. Special instance of cotr2g 13425. (Contributed by RP, 22-Mar-2020.)
dom 𝑅𝐴    &   (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵    &   ran 𝑅𝐶       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 
Theoremcotr3 13427* 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 13428 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 13429 The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
 
Theorem0trrel 13430 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(∅ ∘ ∅) ⊆ ∅
 
5.8.2  Basic properties of closures
 
Theoremcleq1lem 13431 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))
 
Theoremcleq1 13432* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
 
Theoremclsslem 13433* 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 13434 Extend class notation to include the transitive closure symbol.
class t+
 
Syntaxcrtcl 13435 Extend class notation with reflexive-transitive closure.
class t*
 
Definitiondf-trcl 13436* 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 13437* 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 13438* 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 13439* 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 13440 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
 
Theoremcvbtrcl 13441* 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 13442 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))
 
Theoremtrclexlem 13443 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 13444* 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 13445* 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)
 
TheoremtrclubiOLD 13446* Obsolete version of trclubi 13445 as of 26-Mar-2021. (Contributed by RP, 2-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝑅    &   𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅)
 
Theoremtrclubgi 13447* 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))
 
TheoremtrclubgiOLD 13448* Obsolete version of trclubgi 13447 as of 26-Mar-2021. (Contributed by RP, 3-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑅𝑉        {𝑠 ∣ (𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))
 
Theoremtrclub 13449* 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 13450* 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 13451* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})
 
Theorembrintclab 13452* 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 13453* 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 13454* 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 13455* 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 13456* 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 13457 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 13458 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 13459 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉𝑅 ⊆ (t+‘𝑅))
 
Theoremtrclfvcotr 13460 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))
 
Theoremtrclfvlb2 13461 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))
 
Theoremtrclfvlb3 13462 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))
 
Theoremcotrtrclfv 13463 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)
 
Theoremtrclidm 13464 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))
 
Theoremtrclun 13465 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))
 
Theoremtrclfvg 13466 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 13467 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 13468 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))
 
Theoremdmtrclfv 13469 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 13470 Extend class notation to include relation exponentiation.
class 𝑟
 
Definitiondf-relexp 13471* 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 13472 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
 
Theoremrelexp0 13473 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
 
Theoremrelexp0d 13474 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 13475 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
 
Theoremrelexp1g 13476 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟1) = 𝑅)
 
Theoremdfid5 13477 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ (𝑥𝑟1))
 
Theoremdfid6 13478* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ 𝑛 ∈ {1} (𝑥𝑟𝑛))
 
Theoremrelexpsucr 13479 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))
 
Theoremrelexpsucrd 13480 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 13481 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟1) = 𝑅)
 
Theoremrelexpsucnnl 13482 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
 
Theoremrelexpsucl 13483 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))
 
Theoremrelexpsucld 13484 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 13485 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))
 
Theoremrelexpcnvd 13486 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 13487 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅𝑉 → Rel (𝑅𝑟0))
 
Theoremrelexprelg 13488 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 13489 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉 ∧ Rel 𝑅) → Rel (𝑅𝑟𝑁))
 
Theoremrelexpreld 13490 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 13491 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 13492 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 13493 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 13494 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 13495 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 13496 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 13497 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 13498 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 13499 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 13500 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))
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