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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorems2eq2seq 14301 Two length 2 words are equal iff the corresponding symbols are equal. (Contributed by AV, 20-Oct-2018.)
(((𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → (⟨“𝐴𝐵”⟩ = ⟨“𝐶𝐷”⟩ ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Theorems3eqs2s1eq 14302 Two length 3 words are equal iff the corresponding length 2 words and singleton words consisting of their symbols are equal. (Contributed by AV, 4-Jan-2022.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐷𝑉𝐸𝑉𝐹𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (⟨“𝐴𝐵”⟩ = ⟨“𝐷𝐸”⟩ ∧ ⟨“𝐶”⟩ = ⟨“𝐹”⟩)))

Theorems3eq3seq 14303 Two length 3 words are equal iff the corresponding symbols are equal. (Contributed by AV, 4-Jan-2022.)
(((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐷𝑉𝐸𝑉𝐹𝑉)) → (⟨“𝐴𝐵𝐶”⟩ = ⟨“𝐷𝐸𝐹”⟩ ↔ (𝐴 = 𝐷𝐵 = 𝐸𝐶 = 𝐹)))

Theoremswrds2 14304 Extract two adjacent symbols from a word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
((𝑊 ∈ Word 𝐴𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)

Theoremswrds2m 14305 Extract two adjacent symbols from a word in reverse direction. (Contributed by AV, 11-May-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (2...(♯‘𝑊))) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) = ⟨“(𝑊‘(𝑁 − 2))(𝑊‘(𝑁 − 1))”⟩)

Theoremwrdlen2i 14306 Implications of a word of length two. (Contributed by AV, 27-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))

Theoremwrd2pr2op 14307 A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.) (Proof shortened by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = {⟨0, (𝑊‘0)⟩, ⟨1, (𝑊‘1)⟩})

Theoremwrdlen2 14308 A word of length two. (Contributed by AV, 20-Oct-2018.)
((𝑆𝑉𝑇𝑉) → (𝑊 = {⟨0, 𝑆⟩, ⟨1, 𝑇⟩} ↔ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) ∧ ((𝑊‘0) = 𝑆 ∧ (𝑊‘1) = 𝑇))))

Theoremwrdlen2s2 14309 A word of length two as doubleton word. (Contributed by AV, 20-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theoremwrdl2exs2 14310* A word of length two is a doubleton word. (Contributed by AV, 25-Jan-2021.)
((𝑊 ∈ Word 𝑆 ∧ (♯‘𝑊) = 2) → ∃𝑠𝑆𝑡𝑆 𝑊 = ⟨“𝑠𝑡”⟩)

Theorempfx2 14311 A prefix of length two. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix 2) = ⟨“(𝑊‘0)(𝑊‘1)”⟩)

Theoremwrd3tpop 14312 A word of length three 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 14313 A word of length three as length 3 string. (Contributed by AV, 26-Jan-2021.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) → 𝑊 = ⟨“(𝑊‘0)(𝑊‘1)(𝑊‘2)”⟩)

Theoremrepsw2 14314 The "repeated symbol word" of length two. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 2) = ⟨“𝑆𝑆”⟩)

Theoremrepsw3 14315 The "repeated symbol word" of length three. (Contributed by AV, 6-Nov-2018.)
(𝑆𝑉 → (𝑆 repeatS 3) = ⟨“𝑆𝑆𝑆”⟩)

Theoremswrd2lsw 14316 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 14317 Two words of length at least two 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 AV, 12-Oct-2022.)
((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 1 < (♯‘𝑊)) → (𝑊 = 𝑈 ↔ ((♯‘𝑊) = (♯‘𝑈) ∧ ((𝑊 prefix ((♯‘𝑊) − 2)) = (𝑈 prefix ((♯‘𝑊) − 2)) ∧ (𝑊‘((♯‘𝑊) − 2)) = (𝑈‘((♯‘𝑊) − 2)) ∧ (lastS‘𝑊) = (lastS‘𝑈)))))

Theoremccatw2s1ccatws2 14318 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.) (Revised by AV, 29-Jan-2024.)
(𝑊 ∈ Word 𝑉 → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))

Theoremccatw2s1ccatws2OLD 14319 Obsolete version of ccatw2s1ccatws2 14318 as of 29-Jan-2024. 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.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝑋𝑉𝑌𝑉) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) = (𝑊 ++ ⟨“𝑋𝑌”⟩))

Theoremccat2s1fvwALT 14320 Alternate proof of ccat2s1fvw 14000 using words of length 2, see df-s2 14212. 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.) (Revised by AV, 28-Jan-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑊 ∈ Word 𝑉𝐼 ∈ ℕ0𝐼 < (♯‘𝑊)) → (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝐼) = (𝑊𝐼))

Theoremccat2s1fvwALTOLD 14321 Obsolete version of ccat2s1fvwALT 14320 as of 28-Jan-2024. Alternate proof of ccat2s1fvwOLD 14001 using words of length 2, see df-s2 14212. 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 14322* Lemma 1 for wrd2f1tovbij 14326. (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 14323* Lemma 2 for wrd2f1tovbij 14326. (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 14324* Lemma 3 for wrd2f1tovbij 14326. (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 14325* Lemma 4 for wrd2f1tovbij 14326. (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 14326* 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 14327 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 14328* 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 14329* 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 14330* 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 14331 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 14332 Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.)
((𝐴𝑆𝐵𝑇) → (⟨“𝐴”⟩ ∘f 𝑅⟨“𝐵”⟩) = ⟨“(𝐴𝑅𝐵)”⟩)

Theoremofs2 14333 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, an element of its domain or 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 idref 6901.

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 5961.

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 9169, which is a closure relative to a different transitive property, df-tr 5160.

5.8.1  The reflexive and transitive properties of relations

Theoremcoss12d 14334 Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴𝐵)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐴𝐶) ⊆ (𝐵𝐷))

Theoremtrrelssd 14335 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 14336 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 14337 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 14338* Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 14339 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 14339* Two ways of saying a relation is transitive. Special instance of cotr2g 14338. (Contributed by RP, 22-Mar-2020.)
dom 𝑅𝐴    &   (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵    &   ran 𝑅𝐶       ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Theoremcotr3 14340* 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 14341 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 14342 The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)
((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)

Theorem0trrel 14343 The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(∅ ∘ ∅) ⊆ ∅

5.8.2  Basic properties of closures

Theoremcleq1lem 14344 Equality implies bijection. (Contributed by RP, 9-May-2020.)
(𝐴 = 𝐵 → ((𝐴𝐶𝜑) ↔ (𝐵𝐶𝜑)))

Theoremcleq1 14345* Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
(𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})

Theoremclsslem 14346* 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 14347 Extend class notation to include the transitive closure symbol.
class t+

Syntaxcrtcl 14348 Extend class notation with reflexive-transitive closure.
class t*

Definitiondf-trcl 14349* 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 14350* 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 14351* 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 14352* 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 14353 Equality implies bijection. (Contributed by RP, 5-May-2020.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremcvbtrcl 14354* 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 14355 Equality implies bijection. (Contributed by RP, 9-May-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑆𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

Theoremtrclexlem 14356 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 14357* 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 14358* 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 14359* 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 14360* 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 14361* 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 14362* The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉 → (t+‘𝑅) = {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)})

Theorembrintclab 14363* 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 14364* 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 14365* 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 14366* 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 14367* 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 14368 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 14369 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 14370 The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.)
(𝑅𝑉𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcotr 14371 The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅))

Theoremtrclfvlb2 14372 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅𝑅) ⊆ (t+‘𝑅))

Theoremtrclfvlb3 14373 The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.)
(𝑅𝑉 → (𝑅 ∪ (𝑅𝑅)) ⊆ (t+‘𝑅))

Theoremcotrtrclfv 14374 The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.)
((𝑅𝑉 ∧ (𝑅𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅)

Theoremtrclidm 14375 The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.)
(𝑅𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅))

Theoremtrclun 14376 Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.)
((𝑅𝑉𝑆𝑊) → (t+‘(𝑅𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆))))

Theoremtrclfvg 14377 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 14378 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 14379 The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅))

Theoremdmtrclfv 14380 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 14381 Extend class notation to include relation exponentiation.
class 𝑟

Definitiondf-relexp 14382* 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 14383 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))

Theoremrelexp0 14384 A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))

Theoremrelexp0d 14385 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 14386 A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexp1g 14387 A relation composed once is itself. (Contributed by RP, 22-May-2020.)
(𝑅𝑉 → (𝑅𝑟1) = 𝑅)

Theoremdfid5 14388 Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ (𝑥𝑟1))

Theoremdfid6 14389* Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.)
I = (𝑥 ∈ V ↦ 𝑛 ∈ {1} (𝑥𝑟𝑛))

Theoremrelexpsucr 14390 A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = ((𝑅𝑟𝑁) ∘ 𝑅))

Theoremrelexpsucrd 14391 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 14392 A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝑅𝑟1) = 𝑅)

Theoremrelexpsucnnl 14393 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉𝑁 ∈ ℕ) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucl 14394 A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.)
((𝑅𝑉 ∧ Rel 𝑅𝑁 ∈ ℕ0) → (𝑅𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅𝑟𝑁)))

Theoremrelexpsucld 14395 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 14396 Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉) → (𝑅𝑟𝑁) = (𝑅𝑟𝑁))

Theoremrelexpcnvd 14397 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 14398 The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
(𝑅𝑉 → Rel (𝑅𝑟0))

Theoremrelexprelg 14399 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 14400 The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.)
((𝑁 ∈ ℕ0𝑅𝑉 ∧ Rel 𝑅) → Rel (𝑅𝑟𝑁))

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