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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcnviun 40001* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremimaiun1 40002* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremcoiun1 40003* Composition with an indexed union. Proof analgous to that of coiun 6112. (Contributed by RP, 20-Jun-2020.)
( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremelintima 40004* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
(𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)

Theoremintimass 40005* 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 40006* 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 40007* 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 40008* 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 40009* 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 40010* 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 40011* 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 40012* 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 40013* 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 40014 Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremconrel2d 40015 Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐵𝐴) = ∅)

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

Theoremtrrelind 40016 The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑 → (𝑆𝑆) ⊆ 𝑆)    &   (𝜑𝑇 = (𝑅𝑆))       (𝜑 → (𝑇𝑇) ⊆ 𝑇)

Theoremxpintrreld 40017 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremrestrreld 40018 The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅𝐴))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperreldg 40019 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑆 = (dom 𝑅 × ran 𝑅))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Theoremtrficl 40020* The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)
𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremcnvtrrel 40021 The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperrel2dg 40022 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))

20.31.2.2  Reflexive closures

Syntaxcrcl 40023 Extend class notation with reflexive closure.
class r*

Definitiondf-rcl 40024* 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 40025 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 40026 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 40027* 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))

20.31.2.3  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 40028 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 40029 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 40030* 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 14419. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremeltrclrec 40031* 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 40032* 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 40033* 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 40034* 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 14419. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvmptiunrelexplb0d 40035* 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 40036* 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 40037* 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 40038 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 40039 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 40034. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Theoremfvilbd 40040 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

TheoremfvilbdRP 40041 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 40042 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 40043 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 40044 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 40045 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 40046 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))

Theorembrtrclrec 40047* 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 40048* 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 40049* 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 40050* 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 40051* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14420 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))

Theoremrelexp0eq 40052 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 40053* 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 40054 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 40055 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 40056 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))

Theoremcomptiunov2i 40057* 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 40058 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*

Theoremiunrelexpmin1 40059* 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 40060 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 40061 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 40062* 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 40063* 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 40064 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 40065 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟1)↑𝑟1) = (𝑅𝑟1))

Theoremrelexp0idm 40066 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟0)↑𝑟0) = (𝑅𝑟0))

Theoremrelexp0a 40067 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))

Theoremrelexpxpmin 40068 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 40069 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 14416 or when the sum of the powers isn't 1 as shown by relexpaddg 14415. (Contributed by RP, 3-Jun-2020.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))

Theoremiunrelexpuztr 40070* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14422. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = (ℤ𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶𝑅) ∘ (𝐶𝑅)) ⊆ (𝐶𝑅))

20.31.2.4  Transitive closure of a relation

Theoremdftrcl3 40071* 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 40072* 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 40073 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t+‘𝑅))

Theoremtrclfvcom 40074 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))

Theoremcnvtrclfv 40075 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 40076 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
(t+ ∘ t+) = t+

Theoremtrclimalb2 40077 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)

Theorembrtrclfv2 40078* 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 40079 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 40080 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 40081 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 40082 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 40083 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 40084* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14424. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))

Theorembrfvrtrcld 40085* 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 14419. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvrtrcllb0d 40086 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 40087 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 40088 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 40089 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 40090 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
(r* ∘ t+) = t*

Theoremcortrcltrcl 40091 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t+) = t*

Theoremcorclrtrcl 40092 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(r* ∘ t*) = t*

Theoremcotrclrcl 40093 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
(t+ ∘ r*) = t*

Theoremcortrclrcl 40094 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ r*) = t*

Theoremcotrclrtrcl 40095 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t+ ∘ t*) = t*

Theoremcortrclrtrcl 40096 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t*) = t*

Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].

Theoremfrege77d 40097 If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 40292. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege81d 40098 If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 40296. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)

Theoremfrege83d 40099 If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 40298. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅 “ (𝑈𝑉)) ⊆ (𝑈𝑉))       (𝜑𝐵 ∈ (𝑈𝑉))

Theoremfrege96d 40100 If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 40311. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)

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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 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