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Type | Label | Description |
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Statement | ||
Theorem | s7f1o 15001 | A length 7 word with mutually different symbols is a one-to-one function onto the set of the symbols. (Contributed by AV, 2-Aug-2025.) |
⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐷 ∈ 𝑉 ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐴 ≠ 𝐹 ∧ 𝐴 ≠ 𝐺)) ∧ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ (𝐵 ≠ 𝐸 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐺)) ∧ (𝐶 ≠ 𝐷 ∧ (𝐶 ≠ 𝐸 ∧ 𝐶 ≠ 𝐹 ∧ 𝐶 ≠ 𝐺))) ∧ ((𝐷 ≠ 𝐸 ∧ 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐺) ∧ (𝐸 ≠ 𝐹 ∧ 𝐸 ≠ 𝐺 ∧ 𝐹 ≠ 𝐺)))) → (𝐾 = 〈“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”〉 → 𝐾:(0..^7)–1-1-onto→(({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺}))) | ||
Theorem | s3sndisj 15002* | 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 15003* | 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 15004 | 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 15005 | 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 15006 | 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 7165. 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 6132. 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 9765, which is a closure relative to a different transitive property, df-tr 5265. | ||
Theorem | coss12d 15007 | 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 15008 | 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 15009 | The most interesting case of the composition of two Cartesian products. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷)) | ||
Theorem | xpcoidgend 15010 | If two classes are not disjoint, then the composition of their Cartesian product with itself is idempotent. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅) ⇒ ⊢ (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐵)) | ||
Theorem | cotr2g 15011* | Two ways of saying that the composition of two relations is included in a third relation. See its special instance cotr2 15012 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 15012* | Two ways of saying a relation is transitive. Special instance of cotr2g 15011. (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 15013* | 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 15014 | 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 15015 | The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) | ||
Theorem | 0trrel 15016 | The empty class is a transitive relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (∅ ∘ ∅) ⊆ ∅ | ||
Theorem | cleq1lem 15017 | Equality implies bijection. (Contributed by RP, 9-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) | ||
Theorem | cleq1 15018* | 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 15019* | 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 15020 | Extend class notation to include the transitive closure symbol. |
class t+ | ||
Syntax | crtcl 15021 | Extend class notation with reflexive-transitive closure. |
class t* | ||
Definition | df-trcl 15022* | 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 15023* | 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 15024* | 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 15025* | 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 15026 | Equality implies bijection. (Contributed by RP, 5-May-2020.) |
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵))) | ||
Theorem | cvbtrcl 15027* | 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 15028 | 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 15029 | 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 15030* | 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 15031* | 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 15032* | 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 15033* | 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 15034* | 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 15035* | The transitive closure of a relation. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)}) | ||
Theorem | brintclab 15036* | 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 15037* | 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 15038* | 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 15039* | 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 15040* | 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 15041 | 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 15042 | 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 15043 | The transitive closure of a relation has a lower bound. (Contributed by RP, 28-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → 𝑅 ⊆ (t+‘𝑅)) | ||
Theorem | trclfvcotr 15044 | 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 15045 | The transitive closure of a relation has a lower bound. (Contributed by RP, 8-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∘ 𝑅) ⊆ (t+‘𝑅)) | ||
Theorem | trclfvlb3 15046 | 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 15047 | The transitive closure of a transitive relation. (Contributed by RP, 28-Apr-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅) → (t+‘𝑅) = 𝑅) | ||
Theorem | trclidm 15048 | The transitive closure of a relation is idempotent. (Contributed by RP, 29-Apr-2020.) |
⊢ (𝑅 ∈ 𝑉 → (t+‘(t+‘𝑅)) = (t+‘𝑅)) | ||
Theorem | trclun 15049 | 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 15050 | 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 15051 | 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 15052 | The transitive closure of a relation is a relation. (Contributed by RP, 9-May-2020.) |
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → Rel (t+‘𝑅)) | ||
Theorem | dmtrclfv 15053 | 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 15054 | Extend class notation to include relation exponentiation. |
class ↑𝑟 | ||
Definition | df-relexp 15055* | 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 | reldmrelexp 15056 | The domain of the repeated composition of a relation is a relation. (Contributed by AV, 12-Jul-2024.) |
⊢ Rel dom ↑𝑟 | ||
Theorem | relexp0g 15057 | 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 15058 | 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 15059 | 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.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | relexpsucnnr 15060 | 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 15061 | A relation composed once is itself. (Contributed by RP, 22-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | dfid5 15062 | Identity relation is equal to relational exponentiation to the first power. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ (𝑥↑𝑟1)) | ||
Theorem | dfid6 15063* | Identity relation expressed as indexed union of relational powers. (Contributed by RP, 9-Jun-2020.) |
⊢ I = (𝑥 ∈ V ↦ ∪ 𝑛 ∈ {1} (𝑥↑𝑟𝑛)) | ||
Theorem | relexp1d 15064 | A relation composed once is itself. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑅↑𝑟1) = 𝑅) | ||
Theorem | relexpsucnnl 15065 | 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 15066 | 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 | relexpsucr 15067 | 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 15068 | 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.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | ||
Theorem | relexpsucld 15069 | 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.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | ||
Theorem | relexpcnv 15070 | Commutation of converse and relation exponentiation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexpcnvd 15071 | Commutation of converse and relation exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ◡(𝑅↑𝑟𝑁) = (◡𝑅↑𝑟𝑁)) | ||
Theorem | relexp0rel 15072 | The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.) |
⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) | ||
Theorem | relexprelg 15073 | 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 15074 | 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 15075 | 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.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexpnndm 15076 | 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 15077 | 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 15078 | 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 15079 | 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → dom (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpnnrn 15080 | 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 15081 | 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 15082 | 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 15083 | 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ran (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpfld 15084 | 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 15085 | 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.) |
⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ∪ ∪ (𝑅↑𝑟𝑁) ⊆ ∪ ∪ 𝑅) | ||
Theorem | relexpaddnn 15086 | 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))) | ||
Theorem | relexpuzrel 15087 | The exponentiation of a class to an integer not smaller than 2 is a relation. (Contributed by RP, 23-May-2020.) |
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑅 ∈ 𝑉) → Rel (𝑅↑𝑟𝑁)) | ||
Theorem | relexpaddg 15088 | 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 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉 ∧ ((𝑁 + 𝑀) = 1 → Rel 𝑅))) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Theorem | relexpaddd 15089 | Relation composition becomes addition under exponentiation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑀 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) = (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Syntax | crtrcl 15090 | Extend class notation with recursively defined reflexive, transitive closure. |
class t*rec | ||
Definition | df-rtrclrec 15091* | 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 (𝑟↑𝑟𝑛)) | ||
Theorem | rtrclreclem1 15092 | 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.) |
⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅)) | ||
Theorem | dfrtrclrec2 15093* | 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.) |
⊢ (𝜑 → Rel 𝑅) ⇒ ⊢ (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) | ||
Theorem | rtrclreclem2 15094 | 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.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) | ||
Theorem | rtrclreclem3 15095 | 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.) |
⊢ (𝜑 → Rel 𝑅) ⇒ ⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)) | ||
Theorem | rtrclreclem4 15096* | 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.) |
⊢ (𝜑 → Rel 𝑅) ⇒ ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) | ||
Theorem | dfrtrcl2 15097 | 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.) |
⊢ (𝜑 → Rel 𝑅) ⇒ ⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) | ||
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. | ||
Theorem | relexpindlem 15098* | 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.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑆 ∈ 𝑉) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) | ||
Theorem | relexpind 15099* | 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.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑆 ∈ 𝑉) & ⊢ (𝜂 → 𝑋 ∈ 𝑊) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) | ||
Theorem | rtrclind 15100* | 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.) |
⊢ (𝜂 → Rel 𝑅) & ⊢ (𝜂 → 𝑆 ∈ 𝑉) & ⊢ (𝜂 → 𝑋 ∈ 𝑊) & ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) & ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) & ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) & ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) & ⊢ (𝜂 → 𝜒) & ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) ⇒ ⊢ (𝜂 → (𝑆(t*‘𝑅)𝑋 → 𝜏)) |
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