P. 413 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnvssb 41201	Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ◡𝐴 ⊆ ◡𝐵))
Theorem	relnonrel 41202	The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
⊢ (Rel 𝐴 ↔ (𝐴 ∖ ◡◡𝐴) = ∅)
Theorem	cnvnonrel 41203	The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
Theorem	brnonrel 41204	A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) → ¬ 𝑋(𝐴 ∖ ◡◡𝐴)𝑌)
Theorem	dmnonrel 41205	The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	rnnonrel 41206	The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅
Theorem	resnonrel 41207	A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅
Theorem	imanonrel 41208	An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅
Theorem	cononrel1 41209	Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴) ∘ 𝐵) = ∅
Theorem	cononrel2 41210	Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
⊢ (𝐴 ∘ (𝐵 ∖ ◡◡𝐵)) = ∅
20.31.1.11  RP ADDTO: Functions See also idssxp 5959 by Thierry Arnoux.
Theorem	elmapintab 41211*	Two ways to say a set is an element of mapped intersection of a class. Here 𝐹 maps elements of 𝐶 to elements of ∩ {𝑥 ∣ 𝜑} or 𝑥. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}))    &   ⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥))    ⇒   ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸)))
Theorem	fvnonrel 41212	The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
⊢ ((𝐴 ∖ ◡◡𝐴)‘𝑋) = ∅
Theorem	elinlem 41213	Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))
Theorem	elcnvcnvlem 41214	Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝐴 ∈ ◡◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ ( I ‘𝐴) ∈ 𝐵))
20.31.1.12  RP ADDTO: Finite induction (for finite ordinals) Original probably needs new subsection for Relation-related existence theorems.
Theorem	cnvcnvintabd 41215*	Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ◡◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡◡𝑥 ∧ 𝜓)})
20.31.1.13  RP ADDTO: First and second members of an ordered pair
Theorem	elcnvlem 41216	Two ways to say a set is a member of the converse of a class. (Contributed by RP, 19-Aug-2020.)
⊢ 𝐹 = (𝑥 ∈ (V × V) ↦ ⟨(2nd ‘𝑥), (1st ‘𝑥)⟩)    ⇒   ⊢ (𝐴 ∈ ◡𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹‘𝐴) ∈ 𝐵))
Theorem	elcnvintab 41217*	Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)
⊢ (𝐴 ∈ ◡∩ {𝑥 ∣ 𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑 → 𝐴 ∈ ◡𝑥)))
Theorem	cnvintabd 41218*	Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.)
⊢ (𝜑 → ∃𝑥𝜓)    ⇒   ⊢ (𝜑 → ◡∩ {𝑥 ∣ 𝜓} = ∩ {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = ◡𝑥 ∧ 𝜓)})
20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations
Theorem	undmrnresiss 41219*	Two ways of saying the identity relation restricted to the union of the domain and range of a relation is a subset of a relation. Generalization of reflexg 41220. (Contributed by RP, 26-Sep-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥 ∧ 𝑦𝐵𝑦)))
Theorem	reflexg 41220*	Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦)))
Theorem	cnvssco 41221*	A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
⊢ (◡𝐴 ⊆ ◡(𝐵 ∘ 𝐶) ↔ ∀𝑥∀𝑦∃𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧 ∧ 𝑧𝐵𝑦)))
Theorem	refimssco 41222	Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 → ◡𝐴 ⊆ ◡(𝐴 ∘ 𝐴))
20.31.1.15  RP ADDTO: Basic properties of closures
Theorem	cleq2lem 41223	Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))
Theorem	cbvcllem 41224*	Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}
Theorem	clublem 41225*	If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.)
⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝑥 = 𝑌 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ 𝑌)
Theorem	clss2lem 41226*	The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜓)} ⊆ ∩ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜒)})
Theorem	dfid7 41227*	Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
⊢ I = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ ⊤)})
Theorem	mptrcllem 41228*	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 41229	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 41230*	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 41231	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 41232*	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 41233*	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 41234*	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 41235*	The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
⊢ 𝐴 ∈ 𝑉    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ (𝑥 ∘ 𝑥) ⊆ 𝑥)} ∈ V
Theorem	rtrclexi 41236*	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 41237*	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 41238*	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 41239*	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 41240*	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 41241*	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 41242*	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 41243*	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 41244*	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))})
20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures
Theorem	trcleq2lemRP 41245	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)))
20.31.1.17  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 41256 was motivated by a short Michael Penn video.
Theorem	sqrtcvallem1 41246	Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 41256. (Contributed by RP, 17-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))
Theorem	reabsifneg 41247	Alternate expression for the absolute value of a real number. Lemma for sqrtcval 41256. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))
Theorem	reabsifnpos 41248	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 41249	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 41250	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 41251	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 41252	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 41256. See imsqrtval 41259. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem3 41253	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 41256, sqrtcval2 41257, resqrtval 41258, and imsqrtval 41259. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcvallem4 41254	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 41256. See resqrtval 41258. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))
Theorem	sqrtcvallem5 41255	Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 41258 and imsqrtval 41259. (Contributed by RP, 11-May-2024.)
⊢ (𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)
Theorem	sqrtcval 41256	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 14912 and crimi 14913. 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 41257	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 41256. (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 41258	Real part of the complex square root. (Contributed by RP, 18-May-2024.)
⊢ (𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))
Theorem	imsqrtval 41259	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 41260	Example for resqrtval 41258. (Contributed by RP, 21-May-2024.)
⊢ (ℜ‘(√‘(;15 + (i · 8)))) = 4
Theorem	imsqrtvalex 41261	Example for imsqrtval 41259. (Contributed by RP, 21-May-2024.)
⊢ (ℑ‘(√‘(;15 + (i · 8)))) = 1
20.31.2  Additional statements on relations and subclasses
Theorem	al3im 41262	Version of ax-4 1812 for a nested implication. (Contributed by RP, 13-Apr-2020.)
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))
Theorem	intima0 41263*	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 41264*	Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
Theorem	cnviun 41265*	Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵
Theorem	imaiun1 41266*	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 41267*	Composition with an indexed union. Proof analgous to that of coiun 6164. (Contributed by RP, 20-Jun-2020.)
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
Theorem	elintima 41268*	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 41269*	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 41270*	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 41271*	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 41272*	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 41273*	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 41274*	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 41275*	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 41276*	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 41277*	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 41278	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅)
Theorem	conrel2d 41279	Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ◡𝐴 = ∅)    ⇒   ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅)
20.31.2.1  Transitive relations (not to be confused with transitive classes).
Theorem	trrelind 41280	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 41281	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 41282	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 41283	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 41284*	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 41285	The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆)
Theorem	trrelsuperrel2dg 41286	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))
20.31.2.2  Reflexive closures
Syntax	crcl 41287	Extend class notation with reflexive closure.
class r*
Definition	df-rcl 41288*	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 41289	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 41290	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 41291*	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))
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.
Theorem	relexp2 41292	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 41293	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 41294*	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 14778. (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 41295*	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 41296*	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 41297*	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 41298*	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 14778. (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 41299*	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 41300*	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))