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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wrdlen3s3 14301 | 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)”〉) | ||
Theorem | repsw2 14302 | The "repeated symbol word" of length two. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 2) = 〈“𝑆𝑆”〉) | ||
Theorem | repsw3 14303 | The "repeated symbol word" of length three. (Contributed by AV, 6-Nov-2018.) |
⊢ (𝑆 ∈ 𝑉 → (𝑆 repeatS 3) = 〈“𝑆𝑆𝑆”〉) | ||
Theorem | swrd2lsw 14304 | 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)”〉) | ||
Theorem | 2swrd2eqwrdeq 14305 | 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 𝑉 ∧ 𝑈 ∈ 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))))) | ||
Theorem | ccatw2s1ccatws2 14306 | 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”〉)) | ||
Theorem | ccatw2s1ccatws2OLD 14307 | Obsolete version of ccatw2s1ccatws2 14306 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 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉) = (𝑊 ++ 〈“𝑋𝑌”〉)) | ||
Theorem | ccat2s1fvwALT 14308 | Alternate proof of ccat2s1fvw 13988 using words of length 2, see df-s2 14200. 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 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | ccat2s1fvwALTOLD 14309 | Obsolete version of ccat2s1fvwALT 14308 as of 28-Jan-2024. Alternate proof of ccat2s1fvwOLD 13989 using words of length 2, see df-s2 14200. 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 𝑉 ∧ 𝐼 ∈ ℕ0 ∧ 𝐼 < (♯‘𝑊)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑊 ++ 〈“𝑋”〉) ++ 〈“𝑌”〉)‘𝐼) = (𝑊‘𝐼)) | ||
Theorem | wwlktovf 14310* | Lemma 1 for wrd2f1tovbij 14314. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷⟶𝑅 | ||
Theorem | wwlktovf1 14311* | Lemma 2 for wrd2f1tovbij 14314. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ 𝐹:𝐷–1-1→𝑅 | ||
Theorem | wwlktovfo 14312* | Lemma 3 for wrd2f1tovbij 14314. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–onto→𝑅) | ||
Theorem | wwlktovf1o 14313* | Lemma 4 for wrd2f1tovbij 14314. (Contributed by Alexander van der Vekens, 28-Jul-2018.) |
⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} & ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} & ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) ⇒ ⊢ (𝑃 ∈ 𝑉 → 𝐹:𝐷–1-1-onto→𝑅) | ||
Theorem | wrd2f1tovbij 14314* | 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.) |
⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}) | ||
Theorem | eqwrds3 14315 | 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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑊 = 〈“𝐴𝐵𝐶”〉 ↔ ((♯‘𝑊) = 3 ∧ ((𝑊‘0) = 𝐴 ∧ (𝑊‘1) = 𝐵 ∧ (𝑊‘2) = 𝐶)))) | ||
Theorem | wrdl3s3 14316* | A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 3) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | ||
Theorem | s3sndisj 14317* | The singletons consisting of length 3 strings which have distinct third symbols are disjunct. (Contributed by AV, 17-May-2021.) |
⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → Disj 𝑐 ∈ 𝑍 {〈“𝐴𝐵𝑐”〉}) | ||
Theorem | s3iunsndisj 14318* | The union of singletons consisting of length 3 strings which have distinct first and third symbols are disjunct. (Contributed by AV, 17-May-2021.) |
⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){〈“𝑎𝐵𝑐”〉}) | ||
Theorem | ofccat 14319 | Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 28-Sep-2018.) |
⊢ (𝜑 → 𝐸 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐹 ∈ Word 𝑆) & ⊢ (𝜑 → 𝐺 ∈ Word 𝑇) & ⊢ (𝜑 → 𝐻 ∈ Word 𝑇) & ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐺)) & ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐻)) ⇒ ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘f 𝑅(𝐺 ++ 𝐻)) = ((𝐸 ∘f 𝑅𝐺) ++ (𝐹 ∘f 𝑅𝐻))) | ||
Theorem | ofs1 14320 | Letterwise operations on a single letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (〈“𝐴”〉 ∘f 𝑅〈“𝐵”〉) = 〈“(𝐴𝑅𝐵)”〉) | ||
Theorem | ofs2 14321 | Letterwise operations on a double letter word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇)) → (〈“𝐴𝐵”〉 ∘f 𝑅〈“𝐶𝐷”〉) = 〈“(𝐴𝑅𝐶)(𝐵𝑅𝐷)”〉) | ||
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 (𝑅 ∘ 𝑅) ⊆ 𝑅. See cotr 5966. 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 9159, which is a closure relative to a different transitive property, df-tr 5165. | ||
Theorem | coss12d 14322 | Subset deduction for composition of two classes. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → 𝐴 ⊆ 𝐵) & ⊢ (𝜑 → 𝐶 ⊆ 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐷)) | ||
Theorem | trrelssd 14323 | The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → 𝑆 ⊆ 𝑅) & ⊢ (𝜑 → 𝑇 ⊆ 𝑅) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅) | ||
Theorem | xpcogend 14324 | The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷)) | ||
Theorem | xpcoidgend 14325 | If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) | ||
Theorem | cotr2g 14326* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 14327 for the main application. (Contributed by RP, 22-Mar-2020.) |
⊢ dom 𝐵 ⊆ 𝐷 & ⊢ (ran 𝐵 ∩ dom 𝐴) ⊆ 𝐸 & ⊢ ran 𝐴 ⊆ 𝐹 ⇒ ⊢ ((𝐴 ∘ 𝐵) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐷 ∀𝑦 ∈ 𝐸 ∀𝑧 ∈ 𝐹 ((𝑥𝐵𝑦 ∧ 𝑦𝐴𝑧) → 𝑥𝐶𝑧)) | ||
Theorem | cotr2 14327* | Two ways of saying a relation is transitive. Special instance of cotr2g 14326. (Contributed by RP, 22-Mar-2020.) |
⊢ dom 𝑅 ⊆ 𝐴 & ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵 & ⊢ ran 𝑅 ⊆ 𝐶 ⇒ ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | ||
Theorem | cotr3 14328* | Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.) |
⊢ 𝐴 = dom 𝑅 & ⊢ 𝐵 = (𝐴 ∩ 𝐶) & ⊢ 𝐶 = ran 𝑅 ⇒ ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) | ||
Theorem | coemptyd 14329 | Deduction about composition of classes with no relational content in common. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (dom 𝐴 ∩ ran 𝐵) = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | xptrrel 14330 | The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) | ||
Theorem | 0trrel 14331 | The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (∅ ∘ ∅) ⊆ ∅ | ||
Theorem | cleq1lem 14332 | Equality implies bijection. (Contributed by RP, 9-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) | ||
Theorem | cleq1 14333* | Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) | ||
Theorem | clsslem 14334* | The closure of a subclass is a subclass of the closure. (Contributed by RP, 16-May-2020.) |
⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ 𝜑)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ 𝜑)}) | ||
Syntax | ctcl 14335 | Extend class notation to include the transitive closure symbol. |
class t+ | ||
Syntax | crtcl 14336 | Extend class notation with reflexive-transitive closure. |
class t* | ||
Definition | df-trcl 14337* | 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 ∣ (𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) | ||
Definition | df-rtrcl 14338* | 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)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) | ||
Theorem | trcleq1 14339* | Equality of relations implies equality of transitive closures. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 = 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} = ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | ||
Theorem | trclsslem 14340* | The transitive closure (as a relation) of a subclass is a subclass of the transitive closure. (Contributed by RP, 3-May-2020.) |
⊢ (𝑅 ⊆ 𝑆 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ ∩ {𝑟 ∣ (𝑆 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)}) | ||
Theorem | trcleq2lem 14341 | Equality implies bijection. (Contributed by RP, 5-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
Theorem | cvbtrcl 14342* | Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.) |
⊢ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)} | ||
Theorem | trcleq12lem 14343 | Equality implies bijection. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑆 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
Theorem | trclexlem 14344 | Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 5-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ V) | ||
Theorem | trclublem 14345* | 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 → (𝑅 ∪ (dom 𝑅 × ran 𝑅)) ∈ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | ||
Theorem | trclubi 14346* | 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 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (dom 𝑅 × ran 𝑅) | ||
Theorem | trclubgi 14347* | 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 ∣ (𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)) | ||
Theorem | trclub 14348* | 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) → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ (dom 𝑅 × ran 𝑅)) | ||
Theorem | trclubg 14349* | 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 → ∩ {𝑟 ∣ (𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | ||
Theorem | trclfv 14350* | The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | ||
Theorem | brintclab 14351* | Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.) |
⊢ (𝐴∩ {𝑥 ∣ 𝜑}𝐵 ↔ ∀𝑥(𝜑 → 〈𝐴, 𝐵〉 ∈ 𝑥)) | ||
Theorem | brtrclfv 14352* | Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) | ||
Theorem | brcnvtrclfv 14353* | Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) | ||
Theorem | brtrclfvcnv 14354* | Two ways of expressing the transitive closure of the converse of a binary relation. (Contributed by RP, 10-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝐴(t+‘◡𝑅)𝐵 ↔ ∀𝑟((◡𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))) | ||
Theorem | brcnvtrclfvcnv 14355* | Two ways of expressing the transitive closure of the converse of the converse of a binary relation. (Contributed by RP, 10-May-2020.) |
⊢ ((𝑅 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴◡(t+‘◡𝑅)𝐵 ↔ ∀𝑟((◡𝑅 ⊆ 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟) → 𝐵𝑟𝐴))) | ||
Theorem | trclfvss 14356 | 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 ∈ 𝑊 ∧ 𝑅 ⊆ 𝑆) → (t+‘𝑅) ⊆ (t+‘𝑆)) | ||
Theorem | trclfvub 14357 | The transitive closure of a relation has an upper bound. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅))) | ||
Theorem | trclfvlb 14358 | The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) | ||
Theorem | trclfvcotr 14359 | The transitive closure of a relation is a transitive relation. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ (t+‘𝑅)) ⊆ (t+‘𝑅)) | ||
Theorem | trclfvlb2 14360 | The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ 𝑅) ⊆ (t+‘𝑅)) | ||
Theorem | trclfvlb3 14361 | The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ (t+‘𝑅)) | ||
Theorem | cotrtrclfv 14362 | The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅) | ||
Theorem | trclidm 14363 | The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) | ||
Theorem | trclun 14364 | Transitive closure of a union of relations. (Contributed by RP, 5-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (t+‘(𝑅 ∪ 𝑆)) = (t+‘((t+‘𝑅) ∪ (t+‘𝑆)))) | ||
Theorem | trclfvg 14365 | 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) = ∅) | ||
Theorem | trclfvcotrg 14366 | 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) | ||
Theorem | reltrclfv 14367 | The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) | ||
Theorem | dmtrclfv 14368 | The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 9-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅) | ||
Syntax | crelexp 14369 | Extend class notation to include relation exponentiation. |
class ↑𝑟 | ||
Definition | df-relexp 14370* | 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 ↦ (𝑥 ∘ 𝑟)), (𝑧 ∈ V ↦ 𝑟))‘𝑛))) | ||
Theorem | relexp0g 14371 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | ||
Theorem | relexp0 14372 | A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexp0d 14373 | A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexpsucnnr 14374 | A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexp1g 14375 | A relation composed once is itself. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | dfid5 14376 | Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ (𝑥↑𝑟1)) | ||
Theorem | dfid6 14377* | Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑥↑𝑟𝑛)) | ||
Theorem | relexpsucr 14378 | A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexpsucrd 14379 | A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) | ||
Theorem | relexp1d 14380 | A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | relexpsucnnl 14381 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucl 14382 | A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpsucld 14383 | A reduction for relation exponentiation to the left. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) | ||
Theorem | relexpcnv 14384 | Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexpcnvd 14385 | Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁))) | ||
Theorem | relexp0rel 14386 | The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) | ||
Theorem | relexprelg 14387 | 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 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexprel 14388 | The exponentiation of a relation is a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexpreld 14389 | The exponentiation of a relation is a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → Rel (𝑅↑𝑟𝑁))) | ||
Theorem | relexpnndm 14390 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ dom 𝑅) | ||
Theorem | relexpdmg 14391 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relexpdm 14392 | The domain of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpdmd 14393 | 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.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpnnrn 14394 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ran 𝑅) | ||
Theorem | relexprng 14395 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅)) | ||
Theorem | relexprn 14396 | The range of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexprnd 14397 | 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.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpfld 14398 | The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpfldd 14399 | 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.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝑁 ∈ ℕ0 → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅)) | ||
Theorem | relexpaddnn 14400 | Relation composition becomes addition under exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) |
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