P. 437 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43601-43700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dfid7 43601*	Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})
Theorem	mptrcllem 43602*	Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
⊢ (𝑥 ∈ 𝑉 → ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} ∈ V)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜒)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜃)    &   ⊢ (𝑥 ∈ 𝑉 → 𝜏)    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ↔ 𝜃))    &   ⊢ (𝑧 = ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓 ↔ 𝜏))    ⇒   ⊢ (𝑥 ∈ 𝑉 ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥 ∈ 𝑉 ↦ ∩ {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧 ∧ 𝜓)})
Theorem	cotrintab 43603	The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥)    ⇒   ⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑}
Theorem	rclexi 43604*	The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V
Theorem	rtrclexlem 43605	Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑅 ∈ 𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)
Theorem	rtrclex 43606*	The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
⊢ (𝐴 ∈ V ↔ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)
Theorem	trclubgNEW 43607*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))
Theorem	trclubNEW 43608*	If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → Rel 𝑅)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑅 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))
Theorem	trclexi 43609*	The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V
Theorem	rtrclexi 43610*	The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V
Theorem	clrellem 43611*	When the property 𝜓 holds for a relation substituted for 𝑥, then the closure on that property is a relation if the base set is a relation. (Contributed by RP, 30-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝜑 → Rel 𝑋)    &   ⊢ (𝑥 = ◡◡𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → Rel ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)})
Theorem	clcnvlem 43612*	When 𝐴, an upper bound of the closure, exists and certain substitutions hold the converse of the closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ ((𝜑 ∧ 𝑥 = (◡𝑦 ∪ (𝑋 ∖ ◡◡𝑋))) → (𝜒 → 𝜓))    &   ⊢ ((𝜑 ∧ 𝑦 = ◡𝑥) → (𝜓 → 𝜒))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝜃)    ⇒   ⊢ (𝜑 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ 𝜒)})
Theorem	cnvtrucl0 43613*	The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ⊤)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ⊤)})
Theorem	cnvrcl0 43614*	The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})
Theorem	cnvtrcl0 43615*	The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
⊢ (𝑋 ∈ 𝑉 → ◡∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ∩ {𝑦 ∣ (◡𝑋 ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦)})
Theorem	dmtrcl 43616*	The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → dom ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = dom 𝑋)
Theorem	rntrcl 43617*	The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
⊢ (𝑋 ∈ 𝑉 → ran ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} = ran 𝑋)
Theorem	dfrtrcl5 43618*	Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
⊢ t* = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦))})
21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures
Theorem	trcleq2lemRP 43619	Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴) ↔ (𝑅 ⊆ 𝐵 ∧ (𝐵 ∘ 𝐵) ⊆ 𝐵)))
21.36.4.17  Additions for square root; absolute value This is based on the observation that the real and imaginary parts of a complex number can be calculated from the number's absolute and real part and the sign of its imaginary part. Formalization of the formula in sqrtcval 43630 was motivated by a short Michael Penn video.
Theorem	sqrtcvallem1 43620	Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 43630. (Contributed by RP, 17-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))
Theorem	reabsifneg 43621	Alternate expression for the absolute value of a real number. Lemma for sqrtcval 43630. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))
Theorem	reabsifnpos 43622	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 ≤ 0, -𝐴, 𝐴))
Theorem	reabsifpos 43623	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 < 𝐴, 𝐴, -𝐴))
Theorem	reabsifnneg 43624	Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(0 ≤ 𝐴, 𝐴, -𝐴))
Theorem	reabssgn 43625	Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴))
Theorem	sqrtcvallem2 43626	Equivalent to saying that the square of the imaginary component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43630. See imsqrtval 43633. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem3 43627	Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval 43630, sqrtcval2 43631, resqrtval 43632, and imsqrtval 43633. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcvallem4 43628	Equivalent to saying that the square of the real component of the square root of a complex number is a nonnegative real number. Lemma for sqrtcval 43630. See resqrtval 43632. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem5 43629	Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 43632 and imsqrtval 43633. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcval 43630	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 15227 and crimi 15228. This formula can be found in section 3.7.27 of Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (1965, Dover Press). (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (i · (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))))
Theorem	sqrtcval2 43631	Explicit formula for the complex square root in terms of the square root of nonnegative reals. The right side is slightly more compact than sqrtcval 43630. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘𝐴) = ((√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) + (if((ℑ‘𝐴) < 0, -i, i) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)))))
Theorem	resqrtval 43632	Real part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))
Theorem	imsqrtval 43633	Imaginary part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℑ‘(√‘𝐴)) = (if((ℑ‘𝐴) < 0, -1, 1) · (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2))))
Theorem	resqrtvalex 43634	Example for resqrtval 43632. (Contributed by RP, 21-May-2024.)
⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4
Theorem	imsqrtvalex 43635	Example for imsqrtval 43633. (Contributed by RP, 21-May-2024.)
⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1
21.36.5  Additional statements on relations and subclasses
Theorem	al3im 43636	Version of ax-4 1805 for a nested implication. (Contributed by RP, 13-Apr-2020.)
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))
Theorem	intima0 43637*	Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
⊢ ∩ 𝑎 ∈ 𝐴 (𝑎 “ 𝐵) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ 𝐵)}
Theorem	elimaint 43638*	Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	cnviun 43639*	Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵
Theorem	imaiun1 43640*	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 43641*	Composition with an indexed union. Proof analogous to that of coiun 6277. (Contributed by RP, 20-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	elintima 43642*	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 43643*	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 43644*	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 43645*	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 43646*	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 43647*	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 43648*	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 43649*	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 43650*	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 43651*	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 43652	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)
Theorem	conrel2d 43653	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 43654	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 43655	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 43656	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 43657	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 43658*	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 43659	The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
Theorem	trrelsuperrel2dg 43660	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 43661	Extend class notation with reflexive closure.
class r*
Definition	df-rcl 43662*	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 43663	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 43664	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 43665*	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 43666	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 43667	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 43668*	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 15093. (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 43669*	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 43670*	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 43671*	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 43672*	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 15093. (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 43673*	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 43674*	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 43675*	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 43676	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 43677	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 43672. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	fvilbd 43678	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 43679	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 43680	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 43681	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 43682	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 43683	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 43684	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 43685*	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 43686*	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 43687*	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 43688*	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 43689*	Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15092 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 43690	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 43691*	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 43692	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 43693	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 43694	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 43695*	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 43696	The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ r*) = r*
Theorem	iunrelexpmin1 43697*	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 43698	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 43699	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 43700*	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))