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Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremelmapintrab 36901* Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.)
𝐶 ∈ V    &   𝐶𝐵       (𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝐶))))

21.26.1.8  RP ADDTO: Theorems requiring subset and intersection existence

Theoremelinintrab 36902* Two ways of saying a set is an element of the intersection of a class with the intersection of a class. (Contributed by RP, 14-Aug-2020.)
(𝐴𝑉 → (𝐴 {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = (𝐵𝑥) ∧ 𝜑)} ↔ ((∃𝑥𝜑𝐴𝐵) ∧ ∀𝑥(𝜑𝐴𝑥))))

Theoreminintabss 36903* Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)
(𝐴 {𝑥𝜑}) ⊆ {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜑)}

Theoreminintabd 36904* Value of the intersection of class with the intersection of a non-empty class. (Contributed by RP, 13-Aug-2020.)
(𝜑 → ∃𝑥𝜓)       (𝜑 → (𝐴 {𝑥𝜓}) = {𝑤 ∈ 𝒫 𝐴 ∣ ∃𝑥(𝑤 = (𝐴𝑥) ∧ 𝜓)})

Theoremxpinintabd 36905* Value of the intersection of cross-product with the intersection of a non-empty class. (Contributed by RP, 12-Aug-2020.)
(𝜑 → ∃𝑥𝜓)       (𝜑 → ((𝐴 × 𝐵) ∩ {𝑥𝜓}) = {𝑤 ∈ 𝒫 (𝐴 × 𝐵) ∣ ∃𝑥(𝑤 = ((𝐴 × 𝐵) ∩ 𝑥) ∧ 𝜓)})

Theoremrelintabex 36906 If the intersection of a class is a relation, then the class is non-empty. (Contributed by RP, 12-Aug-2020.)
(Rel {𝑥𝜑} → ∃𝑥𝜑)

Theoremelcnvcnvintab 36907* 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 36908* 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 36909 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 36910 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 36911 Subclass theorem for converse. (Contributed by RP, 22-Oct-2020.)
(Rel 𝐴 → (𝐴𝐵𝐴𝐵))

Theoremrelnonrel 36912 The non-relation part of a relation is empty. (Contributed by RP, 22-Oct-2020.)
(Rel 𝐴 ↔ (𝐴𝐴) = ∅)

Theoremcnvnonrel 36913 The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.)
(𝐴𝐴) = ∅

Theorembrnonrel 36914 A non-relation cannot relate any two classes. (Contributed by RP, 23-Oct-2020.)
((𝑋𝑈𝑌𝑉) → ¬ 𝑋(𝐴𝐴)𝑌)

Theoremdmnonrel 36915 The domain of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
dom (𝐴𝐴) = ∅

Theoremrnnonrel 36916 The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
ran (𝐴𝐴) = ∅

Theoremresnonrel 36917 A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) ↾ 𝐵) = ∅

Theoremimanonrel 36918 An image under the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) “ 𝐵) = ∅

Theoremcononrel1 36919 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
((𝐴𝐴) ∘ 𝐵) = ∅

Theoremcononrel2 36920 Composition with the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
(𝐴 ∘ (𝐵𝐵)) = ∅

Theoremelmapintab 36921* 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 36922 The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
((𝐴𝐴)‘𝑋) = ∅

Theoremelinlem 36923 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 36924 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))

21.26.1.11  RP ADDTO: Finite induction (for finite ordinals)

Original probably needs new subsection for Relation-related existence theorems.

Theoremcnvcnvintabd 36925* 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)})

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

Theoremelcnvlem 36926 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 36927* 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 36928* Value of the converse of the intersection of a non-empty class. (Contributed by RP, 20-Aug-2020.)
(𝜑 → ∃𝑥𝜓)       (𝜑 {𝑥𝜓} = {𝑤 ∈ 𝒫 (V × V) ∣ ∃𝑥(𝑤 = 𝑥𝜓)})

21.26.1.13  RP ADDTO: The reflexive and transitive properties of relations

Theoremundmrnresiss 36929* 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 36930. (Contributed by RP, 26-Sep-2020.)
(( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐵 ↔ ∀𝑥𝑦(𝑥𝐴𝑦 → (𝑥𝐵𝑥𝑦𝐵𝑦)))

Theoremreflexg 36930* 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 36931* A condition weaker than reflexivity. (Contributed by RP, 3-Aug-2020.)
(𝐴(𝐵𝐶) ↔ ∀𝑥𝑦𝑧(𝑥𝐴𝑦 → (𝑥𝐶𝑧𝑧𝐵𝑦)))

Theoremrefimssco 36932 Reflexive relations are subsets of their self-composition. (Contributed by RP, 4-Aug-2020.)
(( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴𝐴(𝐴𝐴))

21.26.1.14  RP ADDTO: Basic properties of closures

Theoremcleq2lem 36933 Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
(𝐴 = 𝐵 → (𝜑𝜓))       (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))

Theoremcbvcllem 36934* 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)}

Theoremintabssd 36935* When for each element 𝑦 there is a subset 𝐴 which may substituted for 𝑥 such that 𝑦 satisfying 𝜒 implies 𝑥 satisfies 𝜓 then the intersection of all 𝑥 that satisfy 𝜓 is a subclass the intersection of all 𝑦 that satisfy 𝜒. (Contributed by RP, 17-Oct-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥 = 𝐴) → (𝜒𝜓))    &   (𝜑𝐴𝑦)       (𝜑 {𝑥𝜓} ⊆ {𝑦𝜒})

Theoremclublem 36936* 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 36937* 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 36938* Definition of identity relation as the trivial closure. (Contributed by RP, 26-Jul-2020.)
I = (𝑥 ∈ V ↦ {𝑦 ∣ (𝑥𝑦 ∧ ⊤)})

Theoremmptrcllem 36939* 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 36940 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 36941* The reflexive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑥)} ∈ V

Theoremrtrclexlem 36942 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 36943* 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 36944* 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 36945* 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 36946* The transitive closure of a set exists. (Contributed by RP, 27-Oct-2020.)
𝐴𝑉        {𝑥 ∣ (𝐴𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} ∈ V

Theoremrtrclexi 36947* 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 36948* 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 36949* 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 36950* 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 36951* 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 36952* 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 36953* 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 36954* 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 36955* 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))})

21.26.1.15  RP REPLACE: Definitions and basic properties of transitive closures

Theoremtrcleq2lemRP 36956 Equality implies bijection. (Contributed by RP, 5-May-2020.) (Proof modification is discouraged.)
(𝐴 = 𝐵 → ((𝑅𝐴 ∧ (𝐴𝐴) ⊆ 𝐴) ↔ (𝑅𝐵 ∧ (𝐵𝐵) ⊆ 𝐵)))

21.26.2  Additional statements on relations and subclasses

Theoremal3im 36957 Version of ax-4 1728 for a nested implication. (Contributed by RP, 13-Apr-2020.)
(∀𝑥(𝜑 → (𝜓 → (𝜒𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃))))

Theoremintima0 36958* Two ways of expressing the intersection of images of a class. (Contributed by RP, 13-Apr-2020.)
𝑎𝐴 (𝑎𝐵) = {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)}

Theoremelimaint 36959* Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
(𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)

Theoremcsbcog 36960 Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))

Theoremcnviun 36961* Converse of indexed union. (Contributed by RP, 20-Jun-2020.)
𝑥𝐴 𝐵 = 𝑥𝐴 𝐵

Theoremimaiun1 36962* The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)

Theoremcoiun1 36963* Composition with an indexed union. Proof analgous to that of coiun 5562. (Contributed by RP, 20-Jun-2020.)
( 𝑥𝐶 𝐴𝐵) = 𝑥𝐶 (𝐴𝐵)

Theoremelintima 36964* Element of intersection of images. (Contributed by RP, 13-Apr-2020.)
(𝑦 {𝑥 ∣ ∃𝑎𝐴 𝑥 = (𝑎𝐵)} ↔ ∀𝑎𝐴𝑏𝐵𝑏, 𝑦⟩ ∈ 𝑎)

Theoremintimass 36965* 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 36966* 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 36967* 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 36968* 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 36969* 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 36970* 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 36971* 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 36972* 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 36973* 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 36974 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐴𝐵) = ∅)

Theoremconrel2d 36975 Deduction about composition with a class with no relational content. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑𝐴 = ∅)       (𝜑 → (𝐵𝐴) = ∅)

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

Theoremtrrelind 36976 The intersection of transitive relations is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑 → (𝑆𝑆) ⊆ 𝑆)    &   (𝜑𝑇 = (𝑅𝑆))       (𝜑 → (𝑇𝑇) ⊆ 𝑇)

Theoremxpintrreld 36977 The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremrestrreld 36978 The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 24-Dec-2019.)
(𝜑 → (𝑅𝑅) ⊆ 𝑅)    &   (𝜑𝑆 = (𝑅𝐴))       (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperreldg 36979 Concrete construction of a superclass of relation 𝑅 which is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
(𝜑 → Rel 𝑅)    &   (𝜑𝑆 = (dom 𝑅 × ran 𝑅))       (𝜑 → (𝑅𝑆 ∧ (𝑆𝑆) ⊆ 𝑆))

Theoremtrficl 36980* The class of all transitive relations has the finite intersection property. (Contributed by Richard Penner, 1-Jan-2020.) (Proof shortened by Richard Penner, 3-Jan-2020.)
𝐴 = {𝑧 ∣ (𝑧𝑧) ⊆ 𝑧}       𝑥𝐴𝑦𝐴 (𝑥𝑦) ∈ 𝐴

Theoremcnvtrrel 36981 The converse of a transitive relation is a transitive relation. (Contributed by Richard Penner, 25-Dec-2019.)
((𝑆𝑆) ⊆ 𝑆 ↔ (𝑆𝑆) ⊆ 𝑆)

Theoremtrrelsuperrel2dg 36982 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))

21.26.2.2  Reflexive closures

Syntaxcrcl 36983 Extend class notation with reflexive closure.
class r*

Definitiondf-rcl 36984* 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 36985 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 36986 Reflexive closure of a relation as union of powers of the relation. (Contributed by RP, 6-Jun-2020.)
r* = (𝑥 ∈ V ↦ ((𝑥𝑟0) ∪ (𝑥𝑟1)))

Theoremdfrcl4 36987* 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))

21.26.2.3  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 36988 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 36989 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 36990* 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 the element is a member of that operator value. Generalized from dfrtrclrec2 13645. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))

Theoremeltrclrec 36991* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))

Theoremelrtrclrec 36992* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅𝑟𝑛)))

Theorembriunov2 36993* 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 the two classes are related by that operator value. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))

Theorembrmptiunrelexpd 36994* 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 13645. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))

Theoremfvmptiunrelexplb0d 36995* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb0da 36996* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → Rel 𝑅)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))

Theoremfvmptiunrelexplb1d 36997* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 1 ∈ 𝑁)       (𝜑𝑅 ⊆ (𝐶𝑅))

Theorembrfvid 36998 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

TheorembrfvidRP 36999 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 36994. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))

Theoremfvilbd 37000 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))

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