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Theorem List for Metamath Proof Explorer - 41001-41100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheorembrfvidRP 41001 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 40996. (Contributed by RP, 21-Jul-2020.) (Proof modification is discouraged.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴( I ‘𝑅)𝐵𝐴𝑅𝐵))
 
Theoremfvilbd 41002 A set is a subset of its image under the identity relation. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ ( I ‘𝑅))
 
TheoremfvilbdRP 41003 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 41004 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 41005 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 41006 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 41007 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 41008 A set is a subset of its image under the reflexive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (r*‘𝑅))
 
Theorembrtrclrec 41009* 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 41010* 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 41011* 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 41012* 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 41013* Any particular operator value is the subset of the index union over a set of operator values. Generalized from rtrclreclem1 14644 and rtrclreclem2 . (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))       ((𝑅𝑈𝑁𝑉𝑀𝑁) → (𝑅 𝑀) ⊆ (𝐶𝑅))
 
Theoremrelexp0eq 41014 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 41015* 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 41016 Any positive power of a Cartesian product of non-disjoint sets is itself. (Contributed by RP, 13-Jun-2020.)
(𝑁 ∈ ℕ → ((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) → ((𝐴 × 𝐵)↑𝑟𝑁) = (𝐴 × 𝐵)))
 
Theoremrelexpiidm 41017 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 41018 The relational power of a subset is a subset. (Contributed by RP, 17-Jun-2020.)
(𝜑𝐴𝐵)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝐴𝑟𝑁) ⊆ (𝐵𝑟𝑁))
 
Theoremcomptiunov2i 41019* 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 41020 The reflexive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(r* ∘ r*) = r*
 
Theoremiunrelexpmin1 41021* 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 41022 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 41023 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 41024* 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 41025* 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 41026 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 41027 Repeated raising a relation to the first power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟1)↑𝑟1) = (𝑅𝑟1))
 
Theoremrelexp0idm 41028 Repeated raising a relation to the zeroth power is idempotent. (Contributed by RP, 12-Jun-2020.)
(𝑅𝑉 → ((𝑅𝑟0)↑𝑟0) = (𝑅𝑟0))
 
Theoremrelexp0a 41029 Absorbtion law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
 
Theoremrelexpxpmin 41030 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.)
(((𝐴𝑈𝐵𝑉 ∧ (𝐴𝐵) ≠ ∅) ∧ (𝐼 = if(𝐽 < 𝐾, 𝐽, 𝐾) ∧ 𝐽 ∈ ℕ0𝐾 ∈ ℕ0)) → (((𝐴 × 𝐵)↑𝑟𝐽)↑𝑟𝐾) = ((𝐴 × 𝐵)↑𝑟𝐼))
 
Theoremrelexpaddss 41031 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 14641 or when the sum of the powers isn't 1 as shown by relexpaddg 14640. (Contributed by RP, 3-Jun-2020.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑅𝑉) → ((𝑅𝑟𝑁) ∘ (𝑅𝑟𝑀)) ⊆ (𝑅𝑟(𝑁 + 𝑀)))
 
Theoremiunrelexpuztr 41032* The indexed union of relation exponentiation over upper integers is a transive relation. Generalized from rtrclreclem3 14647. (Contributed by RP, 4-Jun-2020.)
𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟𝑟𝑛))       ((𝑅𝑉𝑁 = (ℤ𝑀) ∧ 𝑀 ∈ ℕ0) → ((𝐶𝑅) ∘ (𝐶𝑅)) ⊆ (𝐶𝑅))
 
20.31.2.4  Transitive closure of a relation
 
Theoremdftrcl3 41033* 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 41034* 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 41035 A set is a subset of its image under the transitive closure. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑𝑅 ⊆ (t+‘𝑅))
 
Theoremtrclfvcom 41036 The transitive closure of a relation commutes with the relation. (Contributed by RP, 18-Jul-2020.)
(𝑅𝑉 → ((t+‘𝑅) ∘ 𝑅) = (𝑅 ∘ (t+‘𝑅)))
 
Theoremcnvtrclfv 41037 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 41038 The transitive closure is idempotent. (Contributed by RP, 16-Jun-2020.)
(t+ ∘ t+) = t+
 
Theoremtrclimalb2 41039 Lower bound for image under a transitive closure. (Contributed by RP, 1-Jul-2020.)
((𝑅𝑉 ∧ (𝑅 “ (𝐴𝐵)) ⊆ 𝐵) → ((t+‘𝑅) “ 𝐴) ⊆ 𝐵)
 
Theorembrtrclfv2 41040* 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 41041 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 41042 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 41043 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 41044 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 41045 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 41046* Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14649. (Contributed by RP, 5-Jun-2020.)
t* = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
 
Theorembrfvrtrcld 41047* 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 14645. (Contributed by RP, 22-Jul-2020.)
(𝜑𝑅 ∈ V)       (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
 
Theoremfvrtrcllb0d 41048 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 41049 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 41050 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 41051 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 41052 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
(r* ∘ t+) = t*
 
Theoremcortrcltrcl 41053 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t+) = t*
 
Theoremcorclrtrcl 41054 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(r* ∘ t*) = t*
 
Theoremcotrclrcl 41055 The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
(t+ ∘ r*) = t*
 
Theoremcortrclrcl 41056 Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
(t* ∘ r*) = t*
 
Theoremcotrclrtrcl 41057 Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
(t+ ∘ t*) = t*
 
Theoremcortrclrtrcl 41058 The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
(t* ∘ t*) = t*
 
20.31.2.5  Adapted from Frege

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

 
Theoremfrege77d 41059 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 41253. (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 41060 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 41257. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅𝑈) ⊆ 𝑈)       (𝜑𝐵𝑈)
 
Theoremfrege83d 41061 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 41259. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐵)    &   (𝜑 → (𝑅 “ (𝑈𝑉)) ⊆ (𝑈𝑉))       (𝜑𝐵 ∈ (𝑈𝑉))
 
Theoremfrege96d 41062 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 41272. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶 ∈ V)    &   (𝜑𝐴(t+‘𝑅)𝐶)    &   (𝜑𝐶𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege87d 41063 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 41263. (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 41064 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 41267. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴𝑅𝐵)       (𝜑𝐴(t+‘𝑅)𝐵)
 
Theoremfrege97d 41065 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 41273. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = ((t+‘𝑅) “ 𝑈))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)
 
Theoremfrege98d 41066 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 41274. (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 41067 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 41278. (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 41068 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 41282. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴𝑅𝐵)       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))
 
Theoremfrege108d 41069 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 41284. (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 41070 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 41285. (Contributed by RP, 15-Jul-2020.)
(𝜑𝑅 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))       (𝜑 → (𝑅𝐴) ⊆ 𝐴)
 
Theoremfrege114d 41071 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 41290. (Contributed by RP, 15-Jul-2020.)
(𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵))       (𝜑 → (𝐴𝑅𝐵𝐴 = 𝐵𝐵𝑅𝐴))
 
Theoremfrege111d 41072 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 41287. (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 41073 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 41298. (Contributed by RP, 15-Jul-2020.)
(𝜑𝐴 = (𝐹𝑋))    &   (𝜑𝐵 = (𝐹𝑋))       (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 
Theoremfrege124d 41074 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 41300. (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 41075 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 41302. (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 41076 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 41305. (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 41077 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 41307. (Contributed by RP, 17-Jul-2020.)
(𝜑𝐹 ∈ V)    &   (𝜑𝐴 = (𝑈 ∪ (((t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))    &   (𝜑 → Fun 𝐹)       (𝜑 → (𝐹𝐴) ⊆ 𝐴)
 
Theoremfrege133d 41078 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 41309. (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))
 
20.31.3  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 3707 for discussion of an example of a class that is not a set.

Numbered propositions from [Frege1879]. ax-frege1 41103, ax-frege2 41104, ax-frege8 41122, ax-frege28 41143, ax-frege31 41147, ax-frege41 41158, frege52 (see ax-frege52a 41170, frege52b 41202, and ax-frege52c 41201 for translations), frege54 (see ax-frege54a 41175, frege54b 41206 and ax-frege54c 41205 for translations) and frege58 (see ax-frege58a 41188, ax-frege58b 41214 and frege58c 41234 for translations) are considered "core" or axioms. However, at least ax-frege8 41122 can be derived from ax-frege1 41103 and ax-frege2 41104, see axfrege8 41120.

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 41170, frege52b 41202, and ax-frege52c 41201. In dffrege69 41245, 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 41086 for a definition in terms of image and subset. In dffrege76 41252, 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 41275, 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 41291, 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 41291 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 41059 for an example.

 
20.31.3.1  _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 848, df-an 400, dfxor4 41079, dfxor5 41080.

Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biconditional (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 2181 (or simpl 486 and simpr 488 and anifp 1072 in the propositional case) and statements similar to cbvalivw 2015, ax-gen 1803, alrimiv 1935, and alrimdv 1937. 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 1833, 𝐴 = V eqv 3429; Some are not B, ¬ ∀𝑥𝑥𝐵, 𝑥¬ 𝑥𝐵 exnal 1834, 𝐵 ⊊ V pssv 4375, 𝐵 ≠ V nev 41083; There are no C, 𝑥¬ 𝑥𝐶, ¬ ∃𝑥𝑥𝐶 alnex 1789, 𝐶 = ∅ eq0 4272; There exist D, ¬ ∀𝑥¬ 𝑥𝐷, 𝑥𝑥𝐷 df-ex 1788, ∅ ⊊ 𝐷 0pss 4373, 𝐷 ≠ ∅ n0 4275.

Notation for relations between expressions also can be written in various ways. All E are P, 𝑥(𝑥𝐸𝑥𝑃), ¬ ∃𝑥(𝑥𝐸 ∧ ¬ 𝑥𝑃) dfss6 3903, 𝐸 = (𝐸𝑃) df-ss 3897, 𝐸𝑃 dfss2 3900; No F are P, 𝑥(𝑥𝐹 → ¬ 𝑥𝑃), ¬ ∃𝑥(𝑥𝐹𝑥𝑃) alinexa 1850, (𝐹𝑃) = ∅ disj1 4379; Some G are not P, ¬ ∀𝑥(𝑥𝐺𝑥𝑃), 𝑥(𝑥𝐺 ∧ ¬ 𝑥𝑃) exanali 1867, (𝐺𝑃) ⊊ 𝐺 nssinpss 4185, ¬ 𝐺𝑃 nss 3977; Some H are P, ¬ ∀𝑥(𝑥𝐻 → ¬ 𝑥𝑃), 𝑥(𝑥𝐻𝑥𝑃) exnalimn 1851, ∅ ⊊ (𝐻𝑃) 0pssin 41084, (𝐻𝑃) ≠ ∅ ndisj 4296.

 
Theoremdfxor4 41079 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 41080 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 41081 Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
((𝜑𝜓𝜒) ↔ (¬ 𝜑 → (¬ 𝜓𝜒)))
 
Theoremdf3an2 41082 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 41083* Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
(𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥𝐴)
 
Theorem0pssin 41084* Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
(∅ ⊊ (𝐴𝐵) ↔ ∃𝑥(𝑥𝐴𝑥𝐵))
 
20.31.3.2  _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.

 
Syntaxwhe 41085 The property of relation 𝑅 being hereditary in class 𝐴.
wff 𝑅 hereditary 𝐴
 
Definitiondf-he 41086 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ 𝐴)
 
Theoremdfhe2 41087 The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ (𝑅𝐴) ⊆ (𝐴 × 𝐴))
 
Theoremdfhe3 41088* The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
 
Theoremheeq12 41089 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐵))
 
Theoremheeq1 41090 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑅 = 𝑆 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
 
Theoremheeq2 41091 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
 
Theoremsbcheg 41092 Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
 
Theoremhess 41093 Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
 
Theoremxphe 41094 Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
(𝐴 × 𝐵) hereditary 𝐵
 
Theorem0he 41095 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
∅ hereditary 𝐴
 
Theorem0heALT 41096 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
∅ hereditary 𝐴
 
Theoremhe0 41097 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
𝐴 hereditary ∅
 
Theoremunhe1 41098 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
 
Theoremsnhesn 41099 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
{⟨𝐴, 𝐴⟩} hereditary {𝐵}
 
Theoremidhe 41100 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
I hereditary 𝐴
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