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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rtrclexi 40001* | 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 40002* | 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 40003* | 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 40004* | 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 40005* | 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 40006* | 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 40007* | 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 40008* | 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 40009* | 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))}) | ||
Theorem | trcleq2lemRP 40010 | 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))) | ||
Theorem | al3im 40011 | Version of ax-4 1810 for a nested implication. (Contributed by RP, 13-Apr-2020.) |
⊢ (∀𝑥(𝜑 → (𝜓 → (𝜒 → 𝜃))) → (∀𝑥𝜑 → (∀𝑥𝜓 → (∀𝑥𝜒 → ∀𝑥𝜃)))) | ||
Theorem | intima0 40012* | 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 40013* | Element of image of intersection. (Contributed by RP, 13-Apr-2020.) |
⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) | ||
Theorem | csbcog 40014 | Distribute proper substitution through a composition of relations. (Contributed by RP, 28-Jun-2020.) |
⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∘ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∘ ⦋𝐴 / 𝑥⦌𝐶)) | ||
Theorem | cnviun 40015* | Converse of indexed union. (Contributed by RP, 20-Jun-2020.) |
⊢ ◡∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ◡𝐵 | ||
Theorem | imaiun1 40016* | 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 40017* | Composition with an indexed union. Proof analgous to that of coiun 6109. (Contributed by RP, 20-Jun-2020.) |
⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) | ||
Theorem | elintima 40018* | 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 40019* | 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 40020* | 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 40021* | 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 40022* | 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 40023* | 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 40024* | 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 40025* | 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 40026* | 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 40027* | 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 40028 | Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ◡𝐴 = ∅) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐵) = ∅) | ||
Theorem | conrel2d 40029 | Deduction about composition with a class with no relational content. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → ◡𝐴 = ∅) ⇒ ⊢ (𝜑 → (𝐵 ∘ 𝐴) = ∅) | ||
Theorem | trrelind 40030 | 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 40031 | The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅) & ⊢ (𝜑 → 𝑆 = (𝑅 ∩ (𝐴 × 𝐵))) ⇒ ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆) | ||
Theorem | restrreld 40032 | 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 40033 | 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 40034* | 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 40035 | The converse of a transitive relation is a transitive relation. (Contributed by RP, 25-Dec-2019.) |
⊢ ((𝑆 ∘ 𝑆) ⊆ 𝑆 ↔ (◡𝑆 ∘ ◡𝑆) ⊆ ◡𝑆) | ||
Theorem | trrelsuperrel2dg 40036 | 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)) | ||
Syntax | crcl 40037 | Extend class notation with reflexive closure. |
class r* | ||
Definition | df-rcl 40038* | 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 40039 | 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 40040 | 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 40041* | 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)) | ||
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 40042 | 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 40043 | 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 40044* | 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 ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) ⇒ ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))) | ||
Theorem | eltrclrec 40045* | 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 40046* | 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 40047* | 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 40048* | 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 14416. (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 40049* | 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 40050* | 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 40051* | 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 40052 | 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 40053 | 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 40048. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) | ||
Theorem | fvilbd 40054 | 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 40055 | 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 40056 | 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 40057 | 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 40058 | 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 40059 | 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 40060 | 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 40061* | 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 40062* | 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 40063* | 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 40064* | 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 40065* | Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14417 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 40066 | 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 40067* | 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 40068 | Any positive power of a cross 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 40069 | 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 40070 | 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 40071* | 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 40072 | The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
⊢ (r* ∘ r*) = r* | ||
Theorem | iunrelexpmin1 40073* | 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 40074 | 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 40075 | 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 40076* | 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 40077* | 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 40078 | 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 40079 | 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 40080 | 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 40081 | Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝐴↑𝑟𝑁)↑𝑟0) ⊆ (𝐴↑𝑟0)) | ||
Theorem | relexpxpmin 40082 | The composition of powers of a cross-product of non-disjoint sets is the cross 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 40083 | 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 14413 or when the sum of the powers isn't 1 as shown by relexpaddg 14412. (Contributed by RP, 3-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Theorem | iunrelexpuztr 40084* | The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14419. (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)) | ||
Theorem | dftrcl3 40085* | 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 40086* | 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 40087 | 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 40088 | 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 40089 | 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 40090 | The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.) |
⊢ (t+ ∘ t+) = t+ | ||
Theorem | trclimalb2 40091 | 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 40092* | 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 40093 | 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 40094 | 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 40095 | 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 40096 | 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 40097 | 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 40098* | Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14421. (Contributed by RP, 5-Jun-2020.) |
⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | ||
Theorem | brfvrtrcld 40099* | 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 14416. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) | ||
Theorem | fvrtrcllb0d 40100 | 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)) |
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