![]() |
Metamath
Proof Explorer Theorem List (p. 432 of 483) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30741) |
![]() (30742-32264) |
![]() (32265-48238) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | trrelsuperrel2dg 43101 | 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 43102 | Extend class notation with reflexive closure. |
class r* | ||
Definition | df-rcl 43103* | 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 43104 | 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 43105 | 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 43106* | 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 43107 | 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 43108 | 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 43109* | 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 15038. (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 43110* | 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 43111* | 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 43112* | 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 43113* | 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 15038. (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 43114* | 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 43115* | 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 43116* | 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 43117 | 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 43118 | 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 43113. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵)) | ||
Theorem | fvilbd 43119 | 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 43120 | 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 43121 | 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 43122 | 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 43123 | 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 43124 | 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 43125 | 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 43126* | 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 43127* | 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 43128* | 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 43129* | 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 43130* | Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15037 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 43131 | 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 43132* | 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 43133 | 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 43134 | 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 43135 | 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 43136* | 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 43137 | The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
⊢ (r* ∘ r*) = r* | ||
Theorem | iunrelexpmin1 43138* | 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 43139 | 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 43140 | 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 43141* | 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 43142* | 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 43143 | 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 43144 | 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 43145 | 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 43146 | Absorption law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝐴↑𝑟𝑁)↑𝑟0) ⊆ (𝐴↑𝑟0)) | ||
Theorem | relexpxpmin 43147 | The composition of powers of a Cartesian product of non-disjoint sets is the Cartesian 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 43148 | 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 15034 or when the sum of the powers isn't 1 as shown by relexpaddg 15033. (Contributed by RP, 3-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀))) | ||
Theorem | iunrelexpuztr 43149* | The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 15040. (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 43150* | 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 43151* | 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 43152 | 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 43153 | 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 43154 | 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 43155 | The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.) |
⊢ (t+ ∘ t+) = t+ | ||
Theorem | trclimalb2 43156 | 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 43157* | 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 43158 | 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 43159 | 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 43160 | 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 43161 | 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 43162 | 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 43163* | Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15042. (Contributed by RP, 5-Jun-2020.) |
⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) | ||
Theorem | brfvrtrcld 43164* | 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 15038. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵)) | ||
Theorem | fvrtrcllb0d 43165 | 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)) | ||
Theorem | fvrtrcllb0da 43166 | A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → Rel 𝑅) & ⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*‘𝑅)) | ||
Theorem | fvrtrcllb1d 43167 | A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) ⇒ ⊢ (𝜑 → 𝑅 ⊆ (t*‘𝑅)) | ||
Theorem | dfrtrcl4 43168 | Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.) |
⊢ t* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟))) | ||
Theorem | corcltrcl 43169 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.) |
⊢ (r* ∘ t+) = t* | ||
Theorem | cortrcltrcl 43170 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ t+) = t* | ||
Theorem | corclrtrcl 43171 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (r* ∘ t*) = t* | ||
Theorem | cotrclrcl 43172 | The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.) |
⊢ (t+ ∘ r*) = t* | ||
Theorem | cortrclrcl 43173 | Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ r*) = t* | ||
Theorem | cotrclrtrcl 43174 | Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.) |
⊢ (t+ ∘ t*) = t* | ||
Theorem | cortrclrtrcl 43175 | The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.) |
⊢ (t* ∘ t*) = t* | ||
Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879]. | ||
Theorem | frege77d 43176 | If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 43370. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) & ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege81d 43177 | If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 43374. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege83d 43178 | If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 43376. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) & ⊢ (𝜑 → (𝑅 “ (𝑈 ∪ 𝑉)) ⊆ (𝑈 ∪ 𝑉)) ⇒ ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∪ 𝑉)) | ||
Theorem | frege96d 43179 | If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 43389. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege87d 43180 | If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 43380. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶𝑅𝐵) & ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈) & ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑈) | ||
Theorem | frege91d 43181 | If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 43384. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege97d 43182 | If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 43390. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈)) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege98d 43183 | If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 43391. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶) & ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege102d 43184 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 43395. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵) | ||
Theorem | frege106d 43185 | If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 43399. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege108d 43186 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 43401. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege109d 43187 | If 𝐴 contains all elements of 𝑈 and all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 43402. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈))) ⇒ ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege114d 43188 | If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 43407. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵)) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴)) | ||
Theorem | frege111d 43189 | If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 43404. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝑅 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → 𝐶 ∈ V) & ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶)) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴)) | ||
Theorem | frege122d 43190 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 43415. (Contributed by RP, 15-Jul-2020.) |
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋)) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege124d 43191 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 43417. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵)) | ||
Theorem | frege126d 43192 | If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 43419. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋)) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) | ||
Theorem | frege129d 43193 | If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the transitive closure of 𝐹. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 43422. (Contributed by RP, 16-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝐴 ∈ dom 𝐹) & ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴)) & ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐵(t+‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(t+‘𝐹)𝐵)) | ||
Theorem | frege131d 43194 | If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 43424. (Contributed by RP, 17-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈)))) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴) | ||
Theorem | frege133d 43195 | If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 43426. (Contributed by RP, 18-Jul-2020.) |
⊢ (𝜑 → 𝐹 ∈ V) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴) & ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴)) | ||
In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3775 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 43220, ax-frege2 43221, ax-frege8 43239, ax-frege28 43260, ax-frege31 43264, ax-frege41 43275, frege52 (see ax-frege52a 43287, frege52b 43319, and ax-frege52c 43318 for translations), frege54 (see ax-frege54a 43292, frege54b 43323 and ax-frege54c 43322 for translations) and frege58 (see ax-frege58a 43305, ax-frege58b 43331 and frege58c 43351 for translations) are considered "core" or axioms. However, at least ax-frege8 43239 can be derived from ax-frege1 43220 and ax-frege2 43221, see axfrege8 43237. Frege introduced implication, negation and the universal quantifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 43287, frege52b 43319, and ax-frege52c 43318. In dffrege69 43362, Frege introduced 𝑅 hereditary 𝐴 to say that relation 𝑅, when restricted to operate on elements of class 𝐴, will only have elements of class 𝐴 in its domain; see df-he 43203 for a definition in terms of image and subset. In dffrege76 43369, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write 𝑋(t+‘𝑅)𝑌, which requires 𝑅 to also be a set. In dffrege99 43392, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write 𝑋((t+‘𝑅) ∪ I )𝑌, which is a superclass of sets related by the reflexive-transitive relation 𝑋(t*‘𝑅)𝑌. Finally, in dffrege115 43408, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun ◡◡𝑅 (to ignore any non-relational content of the class 𝑅). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html 43408 for details. English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 43176 for an example. | ||
Section 2 introduces the turnstile ⊢ which turns an idea which may be true 𝜑 into an assertion that it does hold true ⊢ 𝜑. Section 5 introduces implication, (𝜑 → 𝜓). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or (¬ 𝜑 → 𝜓), and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 847, df-an 396, dfxor4 43196, dfxor5 43197. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biconditional (𝜑 ↔ 𝜓) or class equality 𝐴 = 𝐵 in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f(𝜑) is interpreted to mean if-(𝜑, 𝜓, 𝜒) where the content of the "function" is specified by the latter two argments or logical equivalent, while g(𝐴) is read as 𝐴 ∈ 𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴 ∈ 𝐺 and 𝐴𝐻𝐵 cannot hold unless 𝐴 is a set. (Also 𝐵.) Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f(𝜑) as if-(𝜑, 𝜓, 𝜒) would result in the translation of ∀𝜑 f (𝜑) as (𝜓 ∧ 𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to ∀𝐴 g(𝐴) being translated as ∀𝑎𝑎 ∈ 𝐺 and so forth. Under this interpreation the text of section 11 gives us sp 2172 (or simpl 482 and simpr 484 and anifp 1070 in the propositional case) and statements similar to cbvalivw 2003, ax-gen 1790, alrimiv 1923, and alrimdv 1925. These last four introduce a generality and have no useful definition in terms of propositional variables. Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, ∀𝑥𝑥 ∈ 𝐴, ¬ ∃𝑥¬ 𝑥 ∈ 𝐴 alex 1821, 𝐴 = V eqv 3480; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1822, 𝐵 ⊊ V pssv 4447, 𝐵 ≠ V nev 43200; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1776, 𝐶 = ∅ eq0 4344; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1775, ∅ ⊊ 𝐷 0pss 4445, 𝐷 ≠ ∅ n0 4347. Notation for relations between expressions also can be written in various ways. All E are P, ∀𝑥(𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐸 ∧ ¬ 𝑥 ∈ 𝑃) dfss6 3969, 𝐸 = (𝐸 ∩ 𝑃) df-ss 3964, 𝐸 ⊆ 𝑃 dfss2 3967; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1838, (𝐹 ∩ 𝑃) = ∅ disj1 4451; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1855, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4257, ¬ 𝐺 ⊆ 𝑃 nss 4044; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1839, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 43201, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4368. | ||
Theorem | dfxor4 43196 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓))) | ||
Theorem | dfxor5 43197 | Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.) |
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓))) | ||
Theorem | df3or2 43198 | Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓 → 𝜒))) | ||
Theorem | df3an2 43199 | Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.) |
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒))) | ||
Theorem | nev 43200* | Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.) |
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |