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Theorem List for Metamath Proof Explorer - 40201-40300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrelintabex 40201 If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.)
(Rel {𝑥𝜑} → ∃𝑥𝜑)

Theoremelcnvcnvintab 40202* Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
(𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))

Theoremrelintab 40203* Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)
(Rel {𝑥𝜑} → {𝑥𝜑} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜑)})

Theoremnonrel 40204 A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.)
(𝐴𝐴) = (𝐴 ∖ (V × V))

Theoremelnonrel 40205 Only an ordered pair where not both entries are sets could be an element of the non-relation part of class. (Contributed by RP, 25-Oct-2020.)
(⟨𝑋, 𝑌⟩ ∈ (𝐴𝐴) ↔ (∅ ∈ 𝐴 ∧ ¬ (𝑋 ∈ V ∧ 𝑌 ∈ V)))

Theoremcnvssb 40206 Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
(Rel 𝐴 → (𝐴𝐵𝐴𝐵))

Theoremrelnonrel 40207 The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
(Rel 𝐴 ↔ (𝐴𝐴) = ∅)

Theoremcnvnonrel 40208 The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
(𝐴𝐴) = ∅

Theorembrnonrel 40209 A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)

Theoremdmnonrel 40210 The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
dom (𝐴𝐴) = ∅

Theoremrnnonrel 40211 The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
ran (𝐴𝐴) = ∅

Theoremresnonrel 40212 A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) ↾ 𝐵) = ∅

Theoremimanonrel 40213 An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) “ 𝐵) = ∅

Theoremcononrel1 40214 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) ∘ 𝐵) = ∅

Theoremcononrel2 40215 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
(𝐴 ∘ (𝐵𝐵)) = ∅

Theoremelmapintab 40216* Two ways to say a set is an element of mapped intersection of a class. Here 𝐹 maps elements of 𝐶 to elements of {𝑥𝜑} or 𝑥. (Contributed by RP, 19-Aug-2020.)
(𝐴𝐵 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ {𝑥𝜑}))    &   (𝐴𝐸 ↔ (𝐴𝐶 ∧ (𝐹𝐴) ∈ 𝑥))       (𝐴𝐵 ↔ (𝐴𝐶 ∧ ∀𝑥(𝜑𝐴𝐸)))

Theoremfvnonrel 40217 The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
((𝐴𝐴)‘𝑋) = ∅

Theoremelinlem 40218 Two ways to say a set is a member of an intersection. (Contributed by RP, 19-Aug-2020.)
(𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ( I ‘𝐴) ∈ 𝐶))

Theoremelcnvcnvlem 40219 Two ways to say a set is a member of the converse of the converse of a class. (Contributed by RP, 20-Aug-2020.)
(𝐴𝐵 ↔ (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.

Theoremcnvcnvintabd 40220* Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
(𝜑 → ∃𝑥𝜓)       (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})

20.31.1.13  RP ADDTO: First and second members of an ordered pair

Theoremelcnvlem 40221 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) ↦ ⟨(2nd𝑥), (1st𝑥)⟩)       (𝐴𝐵 ↔ (𝐴 ∈ (V × V) ∧ (𝐹𝐴) ∈ 𝐵))

Theoremelcnvintab 40222* 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 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))

Theoremcnvintabd 40223* Value of the converse of the intersection of a nonempty class. (Contributed by RP, 20-Aug-2020.)
(𝜑 → ∃𝑥𝜓)       (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})

20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations

Theoremundmrnresiss 40224* 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 40225. (Contributed by RP, 26-Sep-2020.)
(( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))

Theoremreflexg 40225* 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 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥𝑦𝐴𝑦)))

Theoremcnvssco 40226* A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
(𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))

Theoremrefimssco 40227 Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
(( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))

20.31.1.15  RP ADDTO: Basic properties of closures

Theoremcleq2lem 40228 Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))

Theoremcbvcllem 40229* Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
(𝑥 = 𝑦 → (𝜑𝜓))       {𝑥 ∣ (𝑋𝑥𝜑)} = {𝑦 ∣ (𝑋𝑦𝜓)}

Theoremclublem 40230* 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.)
(𝜑𝑌 ∈ V)    &   (𝑥 = 𝑌 → (𝜓𝜒))    &   (𝜑𝑋𝑌)    &   (𝜑𝜒)       (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ 𝑌)

Theoremclss2lem 40231* The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020.)
(𝜑 → (𝜒𝜓))       (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ {𝑥 ∣ (𝑋𝑥𝜒)})

Theoremdfid7 40232* Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})

Theoremmptrcllem 40233* Show two versions of a closure with reflexive properties are equal. (Contributed by RP, 19-Oct-2020.)
(𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} ∈ V)    &   (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} ∈ V)    &   (𝑥𝑉𝜒)    &   (𝑥𝑉𝜃)    &   (𝑥𝑉𝜏)    &   (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (𝜑𝜒))    &   (𝑦 = {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)} → (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦𝜃))    &   (𝑧 = {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))} → (𝜓𝜏))       (𝑥𝑉 {𝑦 ∣ (𝑥𝑦 ∧ (𝜑 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦))}) = (𝑥𝑉 {𝑧 ∣ ((𝑥 ∪ ( I ↾ (dom 𝑥 ∪ ran 𝑥))) ⊆ 𝑧𝜓)})

Theoremcotrintab 40234 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 → (𝑥𝑥) ⊆ 𝑥)       ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}

Theoremrclexi 40235* The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Theoremrtrclexlem 40236 Existence of relation implies existence of union with Cartesian product of domain and range. (Contributed by RP, 1-Nov-2020.)
(𝑅𝑉 → (𝑅 ∪ ((dom 𝑅 ∪ ran 𝑅) × (dom 𝑅 ∪ ran 𝑅))) ∈ V)

Theoremrtrclex 40237* The reflexive-transitive closure of a set exists. (Contributed by RP, 1-Nov-2020.)
(𝐴 ∈ V ↔ {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V)

TheoremtrclubgNEW 40238* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (𝑅 ∪ (dom 𝑅 × ran 𝑅)))

TheoremtrclubNEW 40239* If a relation exists then the transitive closure has an upper bound. (Contributed by RP, 24-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑 → Rel 𝑅)       (𝜑 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ⊆ (dom 𝑅 × ran 𝑅))

Theoremtrclexi 40240* The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V

Theoremrtrclexi 40241* The reflexive-transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ ((𝑥𝑥) ⊆ 𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥))} ∈ V

Theoremclrellem 40242* 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.)
(𝜑𝑌 ∈ V)    &   (𝜑 → Rel 𝑋)    &   (𝑥 = 𝑌 → (𝜓𝜒))    &   (𝜑𝑋𝑌)    &   (𝜑𝜒)       (𝜑 → Rel {𝑥 ∣ (𝑋𝑥𝜓)})

Theoremclcnvlem 40243* 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.)
((𝜑𝑥 = (𝑦 ∪ (𝑋𝑋))) → (𝜒𝜓))    &   ((𝜑𝑦 = 𝑥) → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝜑𝑋𝐴)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝜃)       (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} = {𝑦 ∣ (𝑋𝑦𝜒)})

Theoremcnvtrucl0 40244* The converse of the trivial closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ⊤)} = {𝑦 ∣ (𝑋𝑦 ∧ ⊤)})

Theoremcnvrcl0 40245* The converse of the reflexive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ ( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦)})

Theoremcnvtrcl0 40246* The converse of the transitive closure is equal to the closure of the converse. (Contributed by RP, 18-Oct-2020.)
(𝑋𝑉 {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑋𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)})

Theoremdmtrcl 40247* The domain of the transitive closure is equal to the domain of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → dom {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = dom 𝑋)

Theoremrntrcl 40248* The range of the transitive closure is equal to the range of its base relation. (Contributed by RP, 1-Nov-2020.)
(𝑋𝑉 → ran {𝑥 ∣ (𝑋𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = ran 𝑋)

Theoremdfrtrcl5 40249* Definition of reflexive-transitive closure as a standard closure. (Contributed by RP, 1-Nov-2020.)
t* = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ (( I ↾ (dom 𝑦 ∪ ran 𝑦)) ⊆ 𝑦 ∧ (𝑦𝑦) ⊆ 𝑦))})

20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures

Theoremtrcleq2lemRP 40250 Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 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 40261 was motivated by a short Michael Penn video.

Theoremsqrtcvallem1 40251 Two ways of saying a complex number does not lie on the positive real axis. Lemma for sqrtcval 40261. (Contributed by RP, 17-May-2024.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (((ℑ‘𝐴) = 0 → (ℜ‘𝐴) ≤ 0) ↔ ¬ 𝐴 ∈ ℝ+))

Theoremreabsifneg 40252 Alternate expression for the absolute value of a real number. Lemma for sqrtcval 40261. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = if(𝐴 < 0, -𝐴, 𝐴))

Theoremreabsifnpos 40253 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))

Theoremreabsifpos 40254 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))

Theoremreabsifnneg 40255 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))

Theoremreabssgn 40256 Alternate expression for the absolute value of a real number. (Contributed by RP, 22-May-2024.)
(𝐴 ∈ ℝ → (abs‘𝐴) = ((sgn‘𝐴) · 𝐴))

Theoremsqrtcvallem2 40257 Equivalent to saying that the square of the imaginary component of the square root of a complex number is a non-negative real number. Lemma for sqrtcval 40261. See imsqrtval 40264. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) − (ℜ‘𝐴)) / 2))

Theoremsqrtcvallem3 40258 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 40261, sqrtcval2 40262, resqrtval 40263, and imsqrtval 40264. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → (√‘(((abs‘𝐴) − (ℜ‘𝐴)) / 2)) ∈ ℝ)

Theoremsqrtcvallem4 40259 Equivalent to saying that the square of the real component of the square root of a complex number is a non-negative real number. Lemma for sqrtcval 40261. See resqrtval 40263. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → 0 ≤ (((abs‘𝐴) + (ℜ‘𝐴)) / 2))

Theoremsqrtcvallem5 40260 Equivalent to saying that the real component of the square root of a complex number is a real number. Lemma for resqrtval 40263 and imsqrtval 40264. (Contributed by RP, 11-May-2024.)
(𝐴 ∈ ℂ → (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)) ∈ ℝ)

Theoremsqrtcval 40261 Explicit formula for the complex square root in terms of the square root of non-negative reals. The right-hand side is decomposed into real and imaginary parts in the format expected by crrei 14551 and crimi 14552. 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))))))

Theoremsqrtcval2 40262 Explicit formula for the complex square root in terms of the square root of non-negative reals. The right side is slightly more compact than sqrtcval 40261. (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)))))

Theoremresqrtval 40263 Real part of the complex square root. (Contributed by RP, 18-May-2024.)
(𝐴 ∈ ℂ → (ℜ‘(√‘𝐴)) = (√‘(((abs‘𝐴) + (ℜ‘𝐴)) / 2)))

Theoremimsqrtval 40264 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))))

Theoremresqrtvalex 40265 Example for resqrtval 40263. (Contributed by RP, 21-May-2024.)
(ℜ‘(√‘(15 + (i · 8)))) = 4

Theoremimsqrtvalex 40266 Example for imsqrtval 40264. (Contributed by RP, 21-May-2024.)
(ℑ‘(√‘(15 + (i · 8)))) = 1

20.31.2  Additional statements on relations and subclasses

Theoremal3im 40267 Version of ax-4 1811 for a nested implication. (Contributed by RP, 13-Apr-2020.)
(∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))

Theoremintima0 40268* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremelimaint 40269* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
(𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)

Theoremcsbcog 40270 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremcnviun 40271* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremimaiun1 40272* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremcoiun1 40273* Composition with an indexed union. Proof analgous to that of coiun 6096. (Contributed by RP, 20-Jun-2020.)
( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremelintima 40274* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
(𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)

Theoremintimass 40275* 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 40276* 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 40277* 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 40278* 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 40279* 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 40280* 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 40281* 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 40282* 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 40283* 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 40284 Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremconrel2d 40285 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 40286 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 40287 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 40288 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 40289 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 40290* 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 40291 The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.)
((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperrel2dg 40292 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 40293 Extend class notation with reflexive closure.
class r*

Definitiondf-rcl 40294* 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 40295 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 40296 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 40297* 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 40298 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 40299 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 40300* 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 14416. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

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