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Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfrcl4 39901* Reflexive closure of a relation as indexed union of powers of the relation. (Contributed by RP, 8-Jun-2020.)
r* = (𝑟 ∈ V ↦ 𝑛 ∈ {0, 1} (𝑟𝑟𝑛))
 
20.30.2.3  Finite relationship composition.

In order for theorems on the transitive closure of a relation to be grouped together before the concept of continuity, we really need an analogue of 𝑟 that works on finite ordinals or finite sets instead of natural numbers.

 
Theoremrelexp2 39902 A set operated on by the relation exponent to the second power is equal to the composition of the set with itself. (Contributed by RP, 1-Jun-2020.)
(𝑅𝑉 → (𝑅𝑟2) = (𝑅𝑅))
 
Theoremrelexpnul 39903 If the domain and range of powers of a relation are disjoint then the relation raised to the sum of those exponents is empty. (Contributed by RP, 1-Jun-2020.)
(((𝑅𝑉 ∧ Rel 𝑅) ∧ (𝑁 ∈ ℕ0𝑀 ∈ ℕ0)) → ((dom (𝑅𝑟𝑁) ∩ ran (𝑅𝑟𝑀)) = ∅ ↔ (𝑅𝑟(𝑁 + 𝑀)) = ∅))
 
Theoremeliunov2 39904* 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 14406. (Contributed by RP, 1-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))
 
Theoremeltrclrec 39905* Membership in the indexed union of relation exponentiation over the natural numbers is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ 𝑋 ∈ (𝑅𝑟𝑛)))
 
Theoremelrtrclrec 39906* Membership in the indexed union of relation exponentiation over the natural numbers (including zero) is equivalent to the existence of at least one number such that the element is a member of that relationship power. (Contributed by RP, 2-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛 ∈ ℕ0 𝑋 ∈ (𝑅𝑟𝑛)))
 
Theorembriunov2 39907* 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 ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
 
Theorembrmptiunrelexpd 39908* 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 14406. (Contributed by RP, 21-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ⊆ ℕ0)       (𝜑 → (𝐴(𝐶𝑅)𝐵 ↔ ∃𝑛𝑁 𝐴(𝑅𝑟𝑛)𝐵))
 
Theoremfvmptiunrelexplb0d 39909* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (𝐶𝑅))
 
Theoremfvmptiunrelexplb0da 39910* If the indexed union ranges over the zeroth power of the relation, then a restriction of the identity relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → Rel 𝑅)    &   (𝜑 → 0 ∈ 𝑁)       (𝜑 → ( I ↾ 𝑅) ⊆ (𝐶𝑅))
 
Theoremfvmptiunrelexplb1d 39911* If the indexed union ranges over the first power of the relation, then the relation is a subset of the appliction of the function to the relation. (Contributed by RP, 22-Jul-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))    &   (𝜑𝑅 ∈ V)    &   (𝜑𝑁 ∈ V)    &   (𝜑 → 1 ∈ 𝑁)       (𝜑𝑅 ⊆ (𝐶𝑅))
 
Theorembrfvid 39912 If two elements are connected by a value of the identity relation, then they are connected via the argument. (Contributed by RP, 21-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
 
TheorembrfvidRP 39913 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 39908. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
 
Theoremfvilbd 39914 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))
 
TheoremfvilbdRP 39915 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))
 
Theorembrfvrcld 39916 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ (𝐴(𝑅𝑟0)𝐵𝐴(𝑅𝑟1)𝐵)))
 
Theorembrfvrcld2 39917 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(r*‘𝑅)𝐵 ↔ ((𝐴 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐵 ∈ (dom 𝑅 ∪ ran 𝑅) ∧ 𝐴 = 𝐵) ∨ 𝐴𝑅𝐵)))
 
Theoremfvrcllb0d 39918 A restriction of the identity relation is a subset of the reflexive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (r*‘𝑅))
 
Theoremfvrcllb0da 39919 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (r*‘𝑅))
 
Theoremfvrcllb1d 39920 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))
 
Theorembrtrclrec 39921* 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 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ 𝑋(𝑅𝑟𝑛)𝑌))
 
Theorembrrtrclrec 39922* 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 (𝑟𝑟𝑛))       (𝑅𝑉 → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛 ∈ ℕ0 𝑋(𝑅𝑟𝑛)𝑌))
 
Theorembriunov2uz 39923* 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 ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
 
Theoremeliunov2uz 39924* 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 ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁 = (ℤ𝑀)) → (𝑋 ∈ (𝐶𝑅) ↔ ∃𝑛𝑁 𝑋 ∈ (𝑅 𝑛)))
 
Theoremov2ssiunov2 39925* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14407 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
 
Theoremrelexp0eq 39926 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.)
((𝐴𝑈𝐵𝑉) → ((dom 𝐴 ∪ ran 𝐴) = (dom 𝐵 ∪ ran 𝐵) ↔ (𝐴𝑟0) = (𝐵𝑟0)))
 
Theoremiunrelexp0 39927* Simplification of zeroth power of indexed union of powers of relations. (Contributed by RP, 19-Jun-2020.)
((𝑅𝑉𝑍 ⊆ ℕ0 ∧ ({0, 1} ∩ 𝑍) ≠ ∅) → ( 𝑥𝑍 (𝑅𝑟𝑥)↑𝑟0) = (𝑅𝑟0))
 
Theoremrelexpxpnnidm 39928 Any positive power of a cross product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
(𝑁 ∈ ℕ → ((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))
 
Theoremrelexpiidm 39929 Any power of any restriction of the identity relation is itself. (Contributed by RP, 12-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → (( I ↾ 𝐴)↑𝑟𝑁) = ( I ↾ 𝐴))
 
Theoremrelexpss1d 39930 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
 
Theoremcomptiunov2i 39931* The composition two indexed unions is sometimes a similar indexed union. (Contributed by RP, 10-Jun-2020.)
𝑋 = (𝑎 ∈ V ↦ 𝑖𝐼 (𝑎 𝑖))    &   𝑌 = (𝑏 ∈ V ↦ 𝑗𝐽 (𝑏 𝑗))    &   𝑍 = (𝑐 ∈ V ↦ 𝑘𝐾 (𝑐 𝑘))    &   𝐼 ∈ V    &   𝐽 ∈ V    &   𝐾 = (𝐼𝐽)    &    𝑘𝐼 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑘𝐽 (𝑑 𝑘) ⊆ 𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖)    &    𝑖𝐼 ( 𝑗𝐽 (𝑑 𝑗) 𝑖) ⊆ 𝑘 ∈ (𝐼𝐽)(𝑑 𝑘)       (𝑋𝑌) = 𝑍
 
Theoremcorclrcl 39932 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*
 
Theoremiunrelexpmin1 39933* 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 ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ) → ∀𝑠((𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
 
Theoremrelexpmulnn 39934 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.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾)) ∧ (𝐽 ∈ ℕ ∧ 𝐾 ∈ ℕ)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
 
Theoremrelexpmulg 39935 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.)
(((𝑅𝑉𝐼 = (𝐽 · 𝐾) ∧ (𝐼 = 0 → 𝐽𝐾)) ∧ (𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
 
Theoremtrclrelexplem 39936* 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 ((𝐷𝑟𝑘)↑𝑟𝑁) ⊆ ( 𝑗 ∈ ℕ (𝐷𝑟𝑗)↑𝑟𝑁))
 
Theoremiunrelexpmin2 39937* 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 ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠𝑅𝑠 ∧ (𝑠𝑠) ⊆ 𝑠) → (𝐶𝑅) ⊆ 𝑠))
 
Theoremrelexp01min 39938 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.)
(((𝑅𝑉𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾)) ∧ (𝐽 ∈ {0, 1} ∧ 𝐾 ∈ {0, 1})) → ((𝑅𝑟𝐽)↑𝑟𝐾) = (𝑅𝑟𝐼))
 
Theoremrelexp1idm 39939 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟1)↑𝑟1) = (𝑅𝑟1))
 
Theoremrelexp0idm 39940 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟0)↑𝑟0) = (𝑅𝑟0))
 
Theoremrelexp0a 39941 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
 
Theoremrelexpxpmin 39942 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.)
(((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → (((𝐴 × 𝐵)↑𝑟𝐽)↑𝑟𝐾) = ((𝐴 × 𝐵)↑𝑟𝐼))
 
Theoremrelexpaddss 39943 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 14403 or when the sum of the powers isn't 1 as shown by relexpaddg 14402. (Contributed by RP, 3-Jun-2020.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
 
Theoremiunrelexpuztr 39944* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14409. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = (ℤ𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶𝑅) ∘ (𝐶𝑅)) ⊆ (𝐶𝑅))
 
20.30.2.4  Transitive closure of a relation
 
Theoremdftrcl3 39945* Transitive closure of a relation, expressed as indexed union of powers of relations. (Contributed by RP, 5-Jun-2020.)
t+ = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ (𝑟𝑟𝑛))
 
Theorembrfvtrcld 39946* 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.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t+‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ 𝐴(𝑅𝑟𝑛)𝐵))
 
Theoremfvtrcllb1d 39947 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t+‘𝑅))
 
Theoremtrclfvcom 39948 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))
 
Theoremcnvtrclfv 39949 The converse of the transitive closure is equal to the transitive closure of the converse relation. (Contributed by RP, 19-Jul-2020.)
(𝑅𝑉(t+‘𝑅) = (t+‘𝑅))
 
Theoremcotrcltrcl 39950 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
(t+ ∘ t+) = t+
 
Theoremtrclimalb2 39951 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
 
Theorembrtrclfv2 39952* Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
 
Theoremtrclfvdecomr 39953 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.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ ((t+‘𝑅) ∘ 𝑅)))
 
Theoremtrclfvdecoml 39954 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.)
(𝑅𝑉 → (t+‘𝑅) = (𝑅 ∪ (𝑅 ∘ (t+‘𝑅))))
 
TheoremdmtrclfvRP 39955 The domain of the transitive closure is equal to the domain of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → dom (t+‘𝑅) = dom 𝑅)
 
TheoremrntrclfvRP 39956 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 19-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
 
Theoremrntrclfv 39957 The range of the transitive closure is equal to the range of the relation. (Contributed by RP, 18-Jul-2020.) (Proof modification is discouraged.)
(𝑅𝑉 → ran (t+‘𝑅) = ran 𝑅)
 
Theoremdfrtrcl3 39958* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14411. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
 
Theorembrfvrtrcld 39959* 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 14406. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
 
Theoremfvrtrcllb0d 39960 A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*‘𝑅))
 
Theoremfvrtrcllb0da 39961 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)    &   (𝜑𝑅 ∈ V)       (𝜑 → ( I ↾ 𝑅) ⊆ (t*‘𝑅))
 
Theoremfvrtrcllb1d 39962 A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t*‘𝑅))
 
Theoremdfrtrcl4 39963 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 ↦ ((𝑟𝑟0) ∪ (t+‘𝑟)))
 
Theoremcorcltrcl 39964 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
(r* ∘ t+) = t*
 
Theoremcortrcltrcl 39965 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t+) = t*
 
Theoremcorclrtrcl 39966 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(r* ∘ t*) = t*
 
Theoremcotrclrcl 39967 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
(t+ ∘ r*) = t*
 
Theoremcortrclrcl 39968 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ r*) = t*
 
Theoremcotrclrtrcl 39969 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t+ ∘ t*) = t*
 
Theoremcortrclrtrcl 39970 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t*) = t*
 
20.30.2.5  Adapted from Frege

Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].

 
Theoremfrege77d 39971 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 40166. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)       (𝜑𝐵𝑈)
 
Theoremfrege81d 39972 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 40170. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)
 
Theoremfrege83d 39973 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 40172. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅 “ (𝑈𝑉)) ⊆ (𝑈𝑉))       (𝜑𝐵 ∈ (𝑈𝑉))
 
Theoremfrege96d 39974 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 40185. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege87d 39975 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 40176. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)    &   (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)
 
Theoremfrege91d 39976 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 40180. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege97d 39977 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 40186. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)
 
Theoremfrege98d 39978 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 40187. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶(t+‘𝑅)𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege102d 39979 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 40191. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege106d 39980 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 40195. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))
 
Theoremfrege108d 39981 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 40197. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵))
 
Theoremfrege109d 39982 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 40198. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)
 
Theoremfrege114d 39983 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 40203. (Contributed by RP, 15-Jul-2020.)
(𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))
 
Theoremfrege111d 39984 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 40200. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑 → (𝐴(t+‘𝑅)𝐶𝐴 = 𝐶))    &   (𝜑𝐶𝑅𝐵)       (𝜑 → (𝐴(t+‘𝑅)𝐵𝐴 = 𝐵𝐵(t+‘𝑅)𝐴))
 
Theoremfrege122d 39985 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 40211. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝐵 = (𝐹𝑋))       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 
Theoremfrege124d 39986 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 40213. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 
Theoremfrege126d 39987 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 40215. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋 ∈ dom 𝐹)    &   (𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))
 
Theoremfrege129d 39988 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 40218. (Contributed by RP, 16-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 ∈ dom 𝐹)    &   (𝜑𝐶 = (𝐹𝐴))    &   (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐵(t+‘𝐹)𝐶𝐵 = 𝐶𝐶(t+‘𝐹)𝐵))
 
Theoremfrege131d 39989 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 40220. (Contributed by RP, 17-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐹𝐴) ⊆ 𝐴)
 
Theoremfrege133d 39990 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 40222. (Contributed by RP, 18-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝑋(t+‘𝐹)𝐴)    &   (𝜑𝑋(t+‘𝐹)𝐵)    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵𝐵(t+‘𝐹)𝐴))
 
20.30.3  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 3770 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 40016, ax-frege2 40017, ax-frege8 40035, ax-frege28 40056, ax-frege31 40060, ax-frege41 40071, frege52 (see ax-frege52a 40083, frege52b 40115, and ax-frege52c 40114 for translations), frege54 (see ax-frege54a 40088, frege54b 40119 and ax-frege54c 40118 for translations) and frege58 (see ax-frege58a 40101, ax-frege58b 40127 and frege58c 40147 for translations) are considered "core" or axioms. However, at least ax-frege8 40035 can be derived from ax-frege1 40016 and ax-frege2 40017, see axfrege8 40033.

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 40083, frege52b 40115, and ax-frege52c 40114. In dffrege69 40158, 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 39999 for a definition in terms of image and subset. In dffrege76 40165, 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 40188, 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 40204, 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 40204 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 39971 for an example.

 
20.30.3.1  _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, (𝜑𝜓). 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 𝜑𝜓), and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 842, df-an 397, dfxor4 39991, dfxor5 39992.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biimplication (𝜑𝜓) 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 𝐴𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴𝐺 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 (𝜓𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to 𝐴 g(𝐴) being translated as 𝑎𝑎𝐺 and so forth.

Under this interpreation the text of section 11 gives us sp 2172 (or simpl 483 and simpr 485 and anifp 1062 in the propositional case) and statements similar to cbvalivw 2005, ax-gen 1787, alrimiv 1919, and alrimdv 1921. 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, 𝑥𝑥𝐴, ¬ ∃𝑥¬ 𝑥𝐴 alex 1817, 𝐴 = V eqv 3503; Some are not B, ¬ ∀𝑥𝑥𝐵, 𝑥¬ 𝑥𝐵 exnal 1818, 𝐵 ⊊ V pssv 4396, 𝐵 ≠ V nev 39995; There are no C, 𝑥¬ 𝑥𝐶, ¬ ∃𝑥𝑥𝐶 alnex 1773, 𝐶 = ∅ eq0 4307; There exist D, ¬ ∀𝑥¬ 𝑥𝐷, 𝑥𝑥𝐷 df-ex 1772, ∅ ⊊ 𝐷 0pss 4394, 𝐷 ≠ ∅ n0 4309.

Notation for relations between expressions also can be written in various ways. All E are P, 𝑥(𝑥𝐸𝑥𝑃), ¬ ∃𝑥(𝑥𝐸 ∧ ¬ 𝑥𝑃) dfss6 3956, 𝐸 = (𝐸𝑃) df-ss 3951, 𝐸𝑃 dfss2 3954; No F are P, 𝑥(𝑥𝐹 → ¬ 𝑥𝑃), ¬ ∃𝑥(𝑥𝐹𝑥𝑃) alinexa 1834, (𝐹𝑃) = ∅ disj1 4399; Some G are not P, ¬ ∀𝑥(𝑥𝐺𝑥𝑃), 𝑥(𝑥𝐺 ∧ ¬ 𝑥𝑃) exanali 1850, (𝐺𝑃) ⊊ 𝐺 nssinpss 4232, ¬ 𝐺𝑃 nss 4028; Some H are P, ¬ ∀𝑥(𝑥𝐻 → ¬ 𝑥𝑃), 𝑥(𝑥𝐻𝑥𝑃) exnalimn 1835, ∅ ⊊ (𝐻𝑃) 0pssin 39996, (𝐻𝑃) ≠ ∅ ndisj 4326.

 
Theoremdfxor4 39991 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((¬ 𝜑𝜓) → ¬ (𝜑 → ¬ 𝜓)))
 
Theoremdfxor5 39992 Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
((𝜑𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑𝜓)))
 
Theoremdf3or2 39993 Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
 
Theoremdf3an2 39994 Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
 
Theoremnev 39995* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
(𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)
 
Theorem0pssin 39996* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
(∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
 
20.30.3.2  _Begriffsschrift_ Notation hints

The statement 𝑅 hereditary 𝐴 means relation 𝑅 is hereditary (in the sense of Frege) in the class 𝐴 or (𝑅𝐴) ⊆ 𝐴. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked.

As Frege was not using the language of classes or sets, this naturally differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

 
Theoremrp-imass 39997 If the 𝑅-image of a class 𝐴 is a subclass of 𝐵, then the restriction of 𝑅 to 𝐴 is a subset of the Cartesian product of 𝐴 and 𝐵. (Contributed by RP, 24-Dec-2019.)
((𝑅𝐴) ⊆ 𝐵 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐵))
 
Syntaxwhe 39998 The property of relation 𝑅 being hereditary in class 𝐴.
wff 𝑅 hereditary 𝐴
 
Definitiondf-he 39999 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
 
Theoremdfhe2 40000 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
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