 Home Metamath Proof ExplorerTheorem List (p. 390 of 437) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28351) Hilbert Space Explorer (28352-29876) Users' Mathboxes (29877-43667)

Theorem List for Metamath Proof Explorer - 38901-39000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmtrcl 38901* The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)

Theoremrntrcl 38902* The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)

Theoremdfrtrcl5 38903* Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})

20.28.1.15  RP REPLACE: Definitions and basic properties of transitive closures

Theoremtrcleq2lemRP 38904 Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

20.28.2  Additional statements on relations and subclasses

Theoremal3im 38905 Version of ax-4 1853 for a nested implication. (Contributed by RP, 13-Apr-2020.)
(∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))

Theoremintima0 38906* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremelimaint 38907* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
(𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)

Theoremcsbcog 38908 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremcnviun 38909* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremimaiun1 38910* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremcoiun1 38911* Composition with an indexed union. Proof analgous to that of coiun 5901. (Contributed by RP, 20-Jun-2020.)
( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremelintima 38912* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
(𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)

Theoremintimass 38913* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
( 𝐴𝐵) ⊆ {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremintimass2 38914* The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
( 𝐴𝐵) ⊆ 𝑥𝐴 (𝑥𝐵)

Theoremintimag 38915* Requirement for the image under the intersection of relations to equal the intersection of the images of those relations. (Contributed by RP, 13-Apr-2020.)
(∀𝑦(∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎) → ( 𝐴𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)})

Theoremintimasn 38916* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
(𝐵𝑉 → ( 𝐴 “ {𝐵}) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎 “ {𝐵})})

Theoremintimasn2 38917* Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
(𝐵𝑉 → ( 𝐴 “ {𝐵}) = 𝑥𝐴 (𝑥 “ {𝐵}))

Theoremss2iundf 38918* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
𝑥𝜑    &   𝑦𝜑    &   𝑦𝑌    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   𝑦𝐶    &   𝑥𝐷    &   𝑦𝐺    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵𝐺)       (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)

Theoremss2iundv 38919* Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵𝐺)       (𝜑 𝑥𝐴 𝐵 𝑦𝐶 𝐷)

Theoremcbviuneq12df 38920* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝑋    &   𝑦𝑌    &   𝑥𝐴    &   𝑦𝐴    &   𝑦𝐵    &   𝑥𝐶    &   𝑦𝐶    &   𝑥𝐷    &   𝑥𝐹    &   𝑦𝐺    &   ((𝜑𝑦𝐶) → 𝑋𝐴)    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐺)    &   ((𝜑𝑦𝐶) → 𝐷 = 𝐹)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremcbviuneq12dv 38921* Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020.)
((𝜑𝑦𝐶) → 𝑋𝐴)    &   ((𝜑𝑥𝐴) → 𝑌𝐶)    &   ((𝜑𝑦𝐶𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ((𝜑𝑥𝐴𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ((𝜑𝑥𝐴) → 𝐵 = 𝐺)    &   ((𝜑𝑦𝐶) → 𝐷 = 𝐹)       (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)

Theoremconrel1d 38922 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremconrel2d 38923 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐵𝐴) = ∅)

20.28.2.1  Transitive relations (not to be confused with transitive classes).

Theoremtrrelind 38924 The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑 → (𝑆𝑆) ⊆ 𝑆)    &   (𝜑𝑇 = (𝑅𝑆))       (𝜑 → (𝑇𝑇) ⊆ 𝑇)

Theoremxpintrreld 38925 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremrestrreld 38926 The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅𝐴))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperreldg 38927 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑆 = (dom 𝑅 × ran 𝑅))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Theoremtrficl 38928* The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremcnvtrrel 38929 The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperrel2dg 38930 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
(𝜑𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

20.28.2.2  Reflexive closures

Syntaxcrcl 38931 Extend class notation with reflexive closure.
class r*

Definitiondf-rcl 38932* Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})

Theoremdfrcl2 38933 Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Theoremdfrcl3 38934 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 38935* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
r* = (𝑟 ∈ V ↦ 𝑛 ∈ {0, 1} (𝑟𝑟𝑛))

20.28.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of 𝑟 that works on finite ordinals or finite sets instead of natural numbers.

Theoremrelexp2 38936 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
(𝑅𝑉 → (𝑅𝑟2) = (𝑅𝑅))

Theoremrelexpnul 38937 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
(((𝑅𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)) → ((dom (𝑅𝑟𝑁) ∩ ran (𝑅𝑟𝑀)) = ∅ ↔ (𝑅𝑟(𝑁 + 𝑀)) = ∅))

Theoremeliunov2 38938* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 14208. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremeltrclrec 38939* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))

Theoremelrtrclrec 38940* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅𝑟𝑛)))

Theorembriunov2 38941* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theorembrmptiunrelexpd 38942* If two elements are connected by an indexed union of relational powers, then they are connected via 𝑛 instances the relation, for some 𝑛. Generalization of dfrtrclrec2 14208. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvmptiunrelexplb0d 38943* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb0da 38944* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → Rel 𝑅)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb1d 38945* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 1 ∈ 𝑁)       (𝜑𝑅 ⊆ (𝐶𝑅))

Theorembrfvid 38946 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

TheorembrfvidRP 38947 If two elements are connected by a value of the identity relation, then they are connected via the argument. This is an example which uses brmptiunrelexpd 38942. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Theoremfvilbd 38948 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

TheoremfvilbdRP 38949 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

Theorembrfvrcld 38950 If two elements are connected by the reflexive closure of a relation, then they are connected via zero or one instances the relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵)))

Theorembrfvrcld2 38951 If two elements are connected by the reflexive closure of a relation, then they are equal or related by relation. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))

Theoremfvrcllb0d 38952 A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))

Theoremfvrcllb0da 38953 A restriction of the identity relation is a subset of the reflexive closure of a relation. (Contributed by RP, 22-Jul-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (r*‘𝑅))

Theoremfvrcllb1d 38954 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))

Theorembrtrclrec 38955* Two classes related by the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅𝑟𝑛)𝑌))

Theorembrrtrclrec 38956* Two classes related by the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the two classes are related by that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))

Theorembriunov2uz 38957* Two classes related by the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the two classes are related by that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theoremeliunov2uz 38958* Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremov2ssiunov2 38959* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14209 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))

Theoremrelexp0eq 38960 The zeroth power of relationships is the same if and only if the union of their domain and ranges is the same. (Contributed by RP, 11-Jun-2020.)
((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))

Theoremiunrelexp0 38961* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))

Theoremrelexpxpnnidm 38962 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
(𝑁 ∈ ℕ → ((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))

Theoremrelexpiidm 38963 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → (( I ↾ 𝐴)↑𝑟𝑁) = ( I ↾ 𝐴))

Theoremrelexpss1d 38964 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))

Theoremcomptiunov2i 38965* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
𝑋 = (𝑎 ∈ V ↦ 𝑖𝐼 (𝑎 𝑖))    &   𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))    &   𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))    &   𝐼 ∈ V    &   𝐽 ∈ V    &   𝐾 = (𝐼𝐽)    &    𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)       (𝑋𝑌) = 𝑍

Theoremcorclrcl 38966 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*

Theoremiunrelexpmin1 38967* The indexed union of relation exponentiation over the natural numbers is the minimum transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

Theoremrelexpmulnn 38968 With exponents limited to the counting numbers, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremrelexpmulg 38969 With ordered exponents, the composition of powers of a relation is the relation raised to the product of exponents. (Contributed by RP, 13-Jun-2020.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremtrclrelexplem 38970* The union of relational powers to positive multiples of 𝑁 is a subset to the transitive closure raised to the power of 𝑁. (Contributed by RP, 15-Jun-2020.)
(𝑁 ∈ ℕ → 𝑘 ∈ ℕ ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))

Theoremiunrelexpmin2 38971* The indexed union of relation exponentiation over the natural numbers (including zero) is the minimum reflexive-transitive relation that includes the relation. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))

Theoremrelexp01min 38972 With exponents limited to 0 and 1, the composition of powers of a relation is the relation raised to the minimum of exponents. (Contributed by RP, 12-Jun-2020.)
(((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))

Theoremrelexp1idm 38973 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟1)↑𝑟1) = (𝑅𝑟1))

Theoremrelexp0idm 38974 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟0)↑𝑟0) = (𝑅𝑟0))

Theoremrelexp0a 38975 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))

Theoremrelexpxpmin 38976 The composition of powers of a cross-product of non-disjoint sets is the cross product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
(((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → (((𝐴 × 𝐵)↑𝑟𝐽)↑𝑟𝐾) = ((𝐴 × 𝐵)↑𝑟𝐼))

Theoremrelexpaddss 38977 The composition of two powers of a relation is a subset of the relation raised to the sum of those exponents. This is equality where 𝑅 is a relation as shown by relexpaddd 14205 or when the sum of the powers isn't 1 as shown by relexpaddg 14204. (Contributed by RP, 3-Jun-2020.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))

Theoremiunrelexpuztr 38978* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14211. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = (ℤ𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶𝑅) ∘ (𝐶𝑅)) ⊆ (𝐶𝑅))

20.28.2.4  Transitive closure of a relation

Theoremdftrcl3 38979* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
t+ = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))

Theorembrfvtrcld 38980* If two elements are connected by the transitive closure of a relation, then they are connected via 𝑛 instances the relation, for some counting number 𝑛. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t+‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvtrcllb1d 38981 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcom 38982 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))

Theoremcnvtrclfv 38983 The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
(𝑅𝑉(t+‘𝑅) = (t+‘𝑅))

Theoremcotrcltrcl 38984 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
(t+ ∘ t+) = t+

Theoremtrclimalb2 38985 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Theorembrtrclfv2 38986* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))

Theoremtrclfvdecomr 38987 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))

Theoremtrclfvdecoml 38988 The transitive closure of a relation may be decomposed into a union of the relation and the composition of the relation with its transitive closure. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ (𝑅 ∘ (t+‘𝑅))))

TheoremdmtrclfvRP 38989 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)

TheoremrntrclfvRP 38990 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

Theoremrntrclfv 38991 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)

Theoremdfrtrcl3 38992* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14213. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))

Theorembrfvrtrcld 38993* If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via 𝑛 instances the relation, for some natural number 𝑛. Similar of dfrtrclrec2 14208. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvrtrcllb0d 38994 A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*‘𝑅))

Theoremfvrtrcllb0da 38995 A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (t*‘𝑅))

Theoremfvrtrcllb1d 38996 A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t*‘𝑅))

Theoremdfrtrcl4 38997 Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ ((𝑟𝑟0) ∪ (t+‘𝑟)))

Theoremcorcltrcl 38998 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
(r* ∘ t+) = t*

Theoremcortrcltrcl 38999 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t+) = t*

Theoremcorclrtrcl 39000 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(r* ∘ t*) = t*

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43667
 Copyright terms: Public domain < Previous  Next >