P. 443 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44201-44300   *Has distinct variable group(s)

Type	Label	Description
Statement
21.36.5.2  Reflexive closures
Syntax	crcl 44201	Extend class notation with reflexive closure.
class r*
Definition	df-rcl 44202*	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 44203	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 44204	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 44205*	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))
21.36.5.3  Finite relationship composition In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of ↑𝑟 that works on finite ordinals or finite sets instead of natural numbers.
Theorem	relexp2 44206	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 44207	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 44208*	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 15066. (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 44209*	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 44210*	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 44211*	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 44212*	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 15066. (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 44213*	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 44214*	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 44215*	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 44216	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 44217	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 44212. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴( I ‘𝑅)𝐵 ↔ 𝐴𝑅𝐵))
Theorem	fvilbd 44218	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 44219	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 44220	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 44221	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 44222	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 44223	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 44224	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 44225*	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 44226*	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 44227*	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 44228*	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 44229*	Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 15065 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 44230	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 44231*	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 44232	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 44233	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 44234	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 44235*	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 44236	The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ r*) = r*
Theorem	iunrelexpmin1 44237*	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 44238	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 44239	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 44240*	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 44241*	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 44242	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 44243	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 44244	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 44245	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 44246	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 44247	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 15062 or when the sum of the powers isn't 1 as shown by relexpaddg 15061. (Contributed by RP, 3-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ (𝑅↑𝑟𝑀)) ⊆ (𝑅↑𝑟(𝑁 + 𝑀)))
Theorem	iunrelexpuztr 44248*	The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 15068. (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))
21.36.5.4  Transitive closure of a relation
Theorem	dftrcl3 44249*	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 44250*	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 44251	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 44252	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 44253	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 44254	The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
⊢ (t+ ∘ t+) = t+
Theorem	trclimalb2 44255	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 44256*	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 44257	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 44258	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 44259	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 44260	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 44261	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 44262*	Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15070. (Contributed by RP, 5-Jun-2020.)
⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
Theorem	brfvrtrcld 44263*	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 15066. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvrtrcllb0d 44264	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 44265	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 44266	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 44267	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 44268	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
⊢ (r* ∘ t+) = t*
Theorem	cortrcltrcl 44269	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t+) = t*
Theorem	corclrtrcl 44270	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ t*) = t*
Theorem	cotrclrcl 44271	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
⊢ (t+ ∘ r*) = t*
Theorem	cortrclrcl 44272	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ r*) = t*
Theorem	cotrclrtrcl 44273	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t+ ∘ t*) = t*
Theorem	cortrclrtrcl 44274	The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t*) = t*
21.36.5.5  Adapted from Frege Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].
Theorem	frege77d 44275	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 44469. (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 44276	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 44473. (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 44277	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 44475. (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 44278	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 44488. (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 44279	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 44479. (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 44280	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 44483. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege97d 44281	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 44489. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege98d 44282	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 44490. (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 44283	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 44494. (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 44284	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 44498. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege108d 44285	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 44500. (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 44286	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 44501. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege114d 44287	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 44506. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))
Theorem	frege111d 44288	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 44503. (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 44289	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 44514. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))    &   ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege124d 44290	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 44516. (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 44291	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 44518. (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 44292	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 44521. (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 44293	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 44523. (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 44294	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 44525. (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))
21.36.6  Propositions from _Begriffsschrift_ 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 3742 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 44319, ax-frege2 44320, ax-frege8 44338, ax-frege28 44359, ax-frege31 44363, ax-frege41 44374, frege52 (see ax-frege52a 44386, frege52b 44418, and ax-frege52c 44417 for translations), frege54 (see ax-frege54a 44391, frege54b 44422 and ax-frege54c 44421 for translations) and frege58 (see ax-frege58a 44404, ax-frege58b 44430 and frege58c 44450 for translations) are considered "core" or axioms. However, at least ax-frege8 44338 can be derived from ax-frege1 44319 and ax-frege2 44320, see axfrege8 44336. 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 44386, frege52b 44418, and ax-frege52c 44417. In dffrege69 44461, 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 44302 for a definition in terms of image and subset. In dffrege76 44468, 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 44491, 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 44507, 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 44507 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 44275 for an example.
21.36.6.1  _Begriffsschrift_ Chapter I 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 859, df-an 400, dfxor4 44295, dfxor5 44296. 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 arguments 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 2217 (or simpl 486 and simpr 488 and anifp 1082 in the propositional case) and statements similar to cbvalivw 2026, ax-gen 1814, alrimiv 1946, and alrimdv 1948. 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 1845, 𝐴 = V eqv 3463; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1846, 𝐵 ⊊ V pssv 4402, 𝐵 ≠ V nev 44299; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1800, 𝐶 = ∅ eq0 4302; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1799, ∅ ⊊ 𝐷 0pss 4400, 𝐷 ≠ ∅ n0 4305. 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 3926, 𝐸 = (𝐸 ∩ 𝑃) dfss2 3922, 𝐸 ⊆ 𝑃 df-ss 3921; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1862, (𝐹 ∩ 𝑃) = ∅ disj1 4405; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1878, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4219, ¬ 𝐺 ⊆ 𝑃 nss 4000; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1863, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 44300, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4322.
Theorem	dfxor4 44295	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 44296	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 44297	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 44298	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 44299*	Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	0pssin 44300*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))