P. 442 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44101-44200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnviun 44101*	Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵
Theorem	imaiun1 44102*	The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)
Theorem	coiun1 44103*	Composition with an indexed union. Proof analogous to that of coiun 6215. (Contributed by RP, 20-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	elintima 44104*	Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)} ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	intimass 44105*	The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}
Theorem	intimass2 44106*	The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.)
⊢ (∩ 𝐴 “ 𝐵) ⊆ ∩ 𝑥 ∈ 𝐴 (𝑥 “ 𝐵)
Theorem	intimag 44107*	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(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ⟨𝑏, 𝑦⟩ ∈ 𝑎 → ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) → (∩ 𝐴 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)})
Theorem	intimasn 44108*	Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})})
Theorem	intimasn2 44109*	Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ 𝑥 ∈ 𝐴 (𝑥 “ {𝐵}))
Theorem	ss2iundf 44110*	Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑦𝑌    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑦𝐺    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	ss2iundv 44111*	Subclass theorem for indexed union. (Contributed by RP, 17-Jul-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐺)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	cbviuneq12df 44112*	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.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑌    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐺    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	cbviuneq12dv 44113*	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.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐹)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑌) → 𝐷 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐺)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐷 = 𝐹)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)
Theorem	conrel1d 44114	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)
Theorem	conrel2d 44115	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)
21.36.5.1  Transitive relations (not to be confused with transitive classes)
Theorem	trrelind 44116	The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)    &   ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆))    ⇒   ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇)
Theorem	xpintrreld 44117	The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))    ⇒   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
Theorem	restrreld 44118	The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)    &   ⊢ (𝜑 → 𝑆 = (𝑅 ↾ 𝐴))    ⇒   ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
Theorem	trrelsuperreldg 44119	Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑆 = (dom 𝑅 × ran 𝑅))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))
Theorem	trficl 44120*	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 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴
Theorem	cnvtrrel 44121	The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
Theorem	trrelsuperrel2dg 44122	Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by RP, 20-Jul-2020.)
⊢ (𝜑 → 𝑆 = (𝑅 ∪ (dom 𝑅 × ran 𝑅)))    ⇒   ⊢ (𝜑 → (𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆))
21.36.5.2  Reflexive closures
Syntax	crcl 44123	Extend class notation with reflexive closure.
class r*
Definition	df-rcl 44124*	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 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
Theorem	dfrcl2 44125	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))
Theorem	dfrcl3 44126	Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
⊢ r* = (𝑥 ∈ V ↦ ((𝑥↑𝑟0) ∪ (𝑥↑𝑟1)))
Theorem	dfrcl4 44127*	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} (𝑟↑𝑟𝑛))
21.36.5.3  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.
Theorem	relexp2 44128	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.)
⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟2) = (𝑅 ∘ 𝑅))
Theorem	relexpnul 44129	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.)
⊢ (((𝑅 ∈ 𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0)) → ((dom (𝑅↑𝑟𝑁) ∩ ran (𝑅↑𝑟𝑀)) = ∅ ↔ (𝑅↑𝑟(𝑁 + 𝑀)) = ∅))
Theorem	eliunov2 44130*	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 15018. (Contributed by RP, 1-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
Theorem	eltrclrec 44131*	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 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅↑𝑟𝑛)))
Theorem	elrtrclrec 44132*	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 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅↑𝑟𝑛)))
Theorem	briunov2 44133*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))
Theorem	brmptiunrelexpd 44134*	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 15018. (Contributed by RP, 21-Jul-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ⊆ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴(𝐶‘𝑅)𝐵 ↔ ∃𝑛 ∈ 𝑁 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvmptiunrelexplb0d 44135*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → 0 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶‘𝑅))
Theorem	fvmptiunrelexplb0da 44136*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 0 ∈ 𝑁)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (𝐶‘𝑅))
Theorem	fvmptiunrelexplb1d 44137*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    &   ⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ V)    &   ⊢ (𝜑 → 1 ∈ 𝑁)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (𝐶‘𝑅))
Theorem	brfvid 44138	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.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	brfvidRP 44139	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 44134. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	fvilbd 44140	A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))
Theorem	fvilbdRP 44141	A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ ( I ‘𝑅))
Theorem	brfvrcld 44142	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.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅↑𝑟0)𝐵 ∨ 𝐴(𝑅↑𝑟1)𝐵)))
Theorem	brfvrcld2 44143	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.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
Theorem	fvrcllb0d 44144	A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))
Theorem	fvrcllb0da 44145	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)    &   ⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (r*‘𝑅))
Theorem	fvrcllb1d 44146	A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (r*‘𝑅))
Theorem	brtrclrec 44147*	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 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅↑𝑟𝑛)𝑌))
Theorem	brrtrclrec 44148*	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 (𝑟↑𝑟𝑛))    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅↑𝑟𝑛)𝑌))
Theorem	briunov2uz 44149*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))
Theorem	eliunov2uz 44150*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
Theorem	ov2ssiunov2 44151*	Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15017 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝑅 ↑ 𝑀) ⊆ (𝐶‘𝑅))
Theorem	relexp0eq 44152	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.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴↑𝑟0) = (𝐵↑𝑟0)))
Theorem	iunrelexp0 44153*	Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → (∪ 𝑥 ∈ 𝑍 (𝑅↑𝑟𝑥)↑𝑟0) = (𝑅↑𝑟0))
Theorem	relexpxpnnidm 44154	Any positive power of a Cartesian product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))
Theorem	relexpiidm 44155	Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (( I ↾ 𝐴)↑𝑟𝑁) = ( I ↾ 𝐴))
Theorem	relexpss1d 44156	The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐴↑𝑟𝑁) ⊆ (𝐵↑𝑟𝑁))
Theorem	comptiunov2i 44157*	The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
⊢ 𝑋 = (𝑎 ∈ V ↦ ∪ 𝑖 ∈ 𝐼 (𝑎 ↑ 𝑖))    &   ⊢ 𝑌 = (𝑏 ∈ V ↦ ∪ 𝑗 ∈ 𝐽 (𝑏 ↑ 𝑗))    &   ⊢ 𝑍 = (𝑐 ∈ V ↦ ∪ 𝑘 ∈ 𝐾 (𝑐 ↑ 𝑘))    &   ⊢ 𝐼 ∈ V    &   ⊢ 𝐽 ∈ V    &   ⊢ 𝐾 = (𝐼 ∪ 𝐽)    &   ⊢ ∪ 𝑘 ∈ 𝐼 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)    &   ⊢ ∪ 𝑘 ∈ 𝐽 (𝑑 ↑ 𝑘) ⊆ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖)    &   ⊢ ∪ 𝑖 ∈ 𝐼 (∪ 𝑗 ∈ 𝐽 (𝑑 ↑ 𝑗) ↑ 𝑖) ⊆ ∪ 𝑘 ∈ (𝐼 ∪ 𝐽)(𝑑 ↑ 𝑘)    ⇒   ⊢ (𝑋 ∘ 𝑌) = 𝑍
Theorem	corclrcl 44158	The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ r*) = r*
Theorem	iunrelexpmin1 44159*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ) → ∀𝑠((𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))
Theorem	relexpmulnn 44160	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.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
Theorem	relexpmulg 44161	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.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽 ≤ 𝐾)) ∧ (𝐽 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0)) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
Theorem	trclrelexplem 44162*	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 ((𝐷↑𝑟𝑘)↑𝑟𝑁) ⊆ (∪ 𝑗 ∈ ℕ (𝐷↑𝑟𝑗)↑𝑟𝑁))
Theorem	iunrelexpmin2 44163*	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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (𝐶‘𝑅) ⊆ 𝑠))
Theorem	relexp01min 44164	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.)
⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅↑𝑟𝐽)↑𝑟𝐾) = (𝑅↑𝑟𝐼))
Theorem	relexp1idm 44165	Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟1)↑𝑟1) = (𝑅↑𝑟1))
Theorem	relexp0idm 44166	Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
⊢ (𝑅 ∈ 𝑉 → ((𝑅↑𝑟0)↑𝑟0) = (𝑅↑𝑟0))
Theorem	relexp0a 44167	Absorption law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝐴↑𝑟𝑁)↑𝑟0) ⊆ (𝐴↑𝑟0))
Theorem	relexpxpmin 44168	The composition of powers of a Cartesian product of non-disjoint sets is the Cartesian product raised to the minimum exponent. (Contributed by RP, 13-Jun-2020.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ0 ∧ 𝐾 ∈ ℕ0)) → (((𝐴 × 𝐵)↑𝑟𝐽)↑𝑟𝐾) = ((𝐴 × 𝐵)↑𝑟𝐼))
Theorem	relexpaddss 44169	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 15014 or when the sum of the powers isn't 1 as shown by relexpaddg 15013. (Contributed by RP, 3-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
Theorem	iunrelexpuztr 44170*	The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 15020. (Contributed by RP, 4-Jun-2020.)
⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟↑𝑟𝑛))    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 = (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶‘𝑅) ∘ (𝐶‘𝑅)) ⊆ (𝐶‘𝑅))
21.36.5.4  Transitive closure of a relation
Theorem	dftrcl3 44171*	Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
⊢ t+ = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ (𝑟↑𝑟𝑛))
Theorem	brfvtrcld 44172*	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.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvtrcllb1d 44173	A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (t+‘𝑅))
Theorem	trclfvcom 44174	The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
⊢ (𝑅 ∈ 𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))
Theorem	cnvtrclfv 44175	The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
⊢ (𝑅 ∈ 𝑉 → ◡(t+‘𝑅) = (t+‘◡𝑅))
Theorem	cotrcltrcl 44176	The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
⊢ (t+ ∘ t+) = t+
Theorem	trclimalb2 44177	Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
⊢ ((𝑅 ∈ 𝑉 ∧ (𝑅 “ (𝐴 ∪ 𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
Theorem	brtrclfv2 44178*	Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑋(t+‘𝑅)𝑌 ↔ 𝑌 ∈ ∩ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
Theorem	trclfvdecomr 44179	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.)
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
Theorem	trclfvdecoml 44180	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.)
⊢ (𝑅 ∈ 𝑉 → (t+‘𝑅) = (𝑅 ∪ (𝑅 ∘ (t+‘𝑅))))
Theorem	dmtrclfvRP 44181	The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝑅 ∈ 𝑉 → dom (t+‘𝑅) = dom 𝑅)
Theorem	rntrclfvRP 44182	The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)
Theorem	rntrclfv 44183	The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝑅 ∈ 𝑉 → ran (t+‘𝑅) = ran 𝑅)
Theorem	dfrtrcl3 44184*	Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15022. (Contributed by RP, 5-Jun-2020.)
⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
Theorem	brfvrtrcld 44185*	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 15018. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvrtrcllb0d 44186	A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*‘𝑅))
Theorem	fvrtrcllb0da 44187	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)    &   ⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*‘𝑅))
Theorem	fvrtrcllb1d 44188	A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (t*‘𝑅))
Theorem	dfrtrcl4 44189	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 ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟)))
Theorem	corcltrcl 44190	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
⊢ (r* ∘ t+) = t*
Theorem	cortrcltrcl 44191	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t+) = t*
Theorem	corclrtrcl 44192	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ t*) = t*
Theorem	cotrclrcl 44193	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
⊢ (t+ ∘ r*) = t*
Theorem	cortrclrcl 44194	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ r*) = t*
Theorem	cotrclrtrcl 44195	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t+ ∘ t*) = t*
Theorem	cortrclrtrcl 44196	The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t*) = t*
21.36.5.5  Adapted from Frege Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].
Theorem	frege77d 44197	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 44391. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈)    &   ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	frege81d 44198	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 44395. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	frege83d 44199	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 44397. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ (𝑈 ∪ 𝑉)) ⊆ (𝑈 ∪ 𝑉))    ⇒   ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∪ 𝑉))
Theorem	frege96d 44200	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 44410. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)