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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhashfzp1 13201 Value of the numeric cardinality of a (possibly empty) integer range. (Contributed by AV, 19-Jun-2021.)
(𝐵 ∈ (ℤ𝐴) → (#‘((𝐴 + 1)...𝐵)) = (𝐵𝐴))

Theoremhashfz0 13202 Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝐵 ∈ ℕ0 → (#‘(0...𝐵)) = (𝐵 + 1))

Theoremhashxplem 13203 Lemma for hashxp 13204. (Contributed by Paul Chapman, 30-Nov-2012.)
𝐵 ∈ Fin       (𝐴 ∈ Fin → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))

Theoremhashxp 13204 The size of the Cartesian product of two finite sets is the product of their sizes. (Contributed by Paul Chapman, 30-Nov-2012.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴 × 𝐵)) = ((#‘𝐴) · (#‘𝐵)))

Theoremhashmap 13205 The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘(𝐴𝑚 𝐵)) = ((#‘𝐴)↑(#‘𝐵)))

Theoremhashpw 13206 The size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by Paul Chapman, 30-Nov-2012.) (Proof shortened by Mario Carneiro, 5-Aug-2014.)
(𝐴 ∈ Fin → (#‘𝒫 𝐴) = (2↑(#‘𝐴)))

Theoremhashfun 13207 A finite set is a function iff it is equinumerous to its domain. (Contributed by Mario Carneiro, 26-Sep-2013.) (Revised by Mario Carneiro, 12-Mar-2015.)
(𝐹 ∈ Fin → (Fun 𝐹 ↔ (#‘𝐹) = (#‘dom 𝐹)))

Theoremhashres 13208 The number of elements of a finite function restricted to a subset of its domain is equal to the number of elements of that subset. (Contributed by AV, 15-Dec-2021.)
((Fun 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘(𝐴𝐵)) = (#‘𝐵))

Theoremhashreshashfun 13209 The number of elements of a finite function expressed by a restriction. (Contributed by AV, 15-Dec-2021.)
((Fun 𝐴𝐴 ∈ Fin ∧ 𝐵 ⊆ dom 𝐴) → (#‘𝐴) = ((#‘(𝐴𝐵)) + (#‘(dom 𝐴𝐵))))

Theoremhashimarn 13210 The size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 equals the size of the function 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) (Proof shortened by AV, 4-May-2021.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 → (#‘(𝐸 “ ran 𝐹)) = (#‘𝐹)))

Theoremhashimarni 13211 If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
((𝐸:dom 𝐸1-1→ran 𝐸𝐸𝑉) → ((𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃 = (𝐸 “ ran 𝐹) ∧ (#‘𝑃) = 𝑁) → (#‘𝐹) = 𝑁))

Theoremresunimafz0 13212 TODO-AV: Revise using 𝐹 ∈ Word dom 𝐼? Formerly part of proof of eupth2lem3 27076: The union of a restriction by an image over an open range of nonnegative integers and a singleton of an ordered pair is a restriction by an image over an interval of nonnegative integers. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)
(𝜑 → Fun 𝐼)    &   (𝜑𝐹:(0..^(#‘𝐹))⟶dom 𝐼)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))       (𝜑 → (𝐼 ↾ (𝐹 “ (0...𝑁))) = ((𝐼 ↾ (𝐹 “ (0..^𝑁))) ∪ {⟨(𝐹𝑁), (𝐼‘(𝐹𝑁))⟩}))

Theoremfnfz0hash 13213 The size of a function on a finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0...𝑁)) → (#‘𝐹) = (𝑁 + 1))

Theoremffz0hash 13214 The size of a function on a finite set of sequential nonnegative integers equals the upper bound of the sequence increased by 1. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0...𝑁)⟶𝐵) → (#‘𝐹) = (𝑁 + 1))

Theoremfnfz0hashnn0 13215 The size of a function on a finite set of sequential nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0...𝑁) → (#‘𝐹) ∈ ℕ0)

Theoremffzo0hash 13216 The size of a function on a half-open range of nonnegative integers. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑁 ∈ ℕ0𝐹 Fn (0..^𝑁)) → (#‘𝐹) = 𝑁)

Theoremfnfzo0hash 13217 The size of a function on a half-open range of nonnegative integers equals the upper bound of this range. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Proof shortened by AV, 11-Apr-2021.)
((𝑁 ∈ ℕ0𝐹:(0..^𝑁)⟶𝐵) → (#‘𝐹) = 𝑁)

Theoremfnfzo0hashnn0 13218 The value of the size function on a half-open range of nonnegative integers is a nonnegative integer. (Contributed by AV, 10-Apr-2021.)
(𝐹 Fn (0..^𝑁) → (#‘𝐹) ∈ ℕ0)

Theoremhashbclem 13219* Lemma for hashbc 13220: inductive step. (Contributed by Mario Carneiro, 13-Jul-2014.)
(𝜑𝐴 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ∀𝑗 ∈ ℤ ((#‘𝐴)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑗}))    &   (𝜑𝐾 ∈ ℤ)       (𝜑 → ((#‘(𝐴 ∪ {𝑧}))C𝐾) = (#‘{𝑥 ∈ 𝒫 (𝐴 ∪ {𝑧}) ∣ (#‘𝑥) = 𝐾}))

Theoremhashbc 13220* The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)
((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))

Theoremhashfacen 13221* The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴𝐵𝐶𝐷) → {𝑓𝑓:𝐴1-1-onto𝐶} ≈ {𝑓𝑓:𝐵1-1-onto𝐷})

Theoremhashf1lem1 13222* Lemma for hashf1 13224. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))    &   (𝜑𝐹:𝐴1-1𝐵)       (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))

Theoremhashf1lem2 13223* Lemma for hashf1 13224. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → ¬ 𝑧𝐴)    &   (𝜑 → ((#‘𝐴) + 1) ≤ (#‘𝐵))       (𝜑 → (#‘{𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵}) = (((#‘𝐵) − (#‘𝐴)) · (#‘{𝑓𝑓:𝐴1-1𝐵})))

Theoremhashf1 13224* The permutation number 𝐴 ∣ ! · ( ∣ 𝐵 ∣ C ∣ 𝐴 ∣ ) = 𝐵 ∣ ! / ( ∣ 𝐵 ∣ − ∣ 𝐴 ∣ )! counts the number of injections from 𝐴 to 𝐵. (Contributed by Mario Carneiro, 21-Jan-2015.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (#‘{𝑓𝑓:𝐴1-1𝐵}) = ((!‘(#‘𝐴)) · ((#‘𝐵)C(#‘𝐴))))

Theoremhashfac 13225* A factorial counts the number of bijections on a finite set. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by Mario Carneiro, 17-Apr-2015.)
(𝐴 ∈ Fin → (#‘{𝑓𝑓:𝐴1-1-onto𝐴}) = (!‘(#‘𝐴)))

Theoremleiso 13226 Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))

Theoremleisorel 13227 Version of isorel 6561 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.)
((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝐷 ↔ (𝐹𝐶) ≤ (𝐹𝐷)))

Theoremfz1isolem 13228* Lemma for fz1iso 13229. (Contributed by Mario Carneiro, 2-Apr-2014.)
𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)    &   𝐵 = (ℕ ∩ ( < “ {((#‘𝐴) + 1)}))    &   𝐶 = (ω ∩ (𝐺‘((#‘𝐴) + 1)))    &   𝑂 = OrdIso(𝑅, 𝐴)       ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))

Theoremfz1iso 13229* Any finite ordered set has an order isometry to a one-based finite sequence. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(#‘𝐴)), 𝐴))

Theoremishashinf 13230* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8158. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(#‘𝑥) = 𝑛)

Theoremseqcoll 13231* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 2-Apr-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝑁 ∈ (1...(#‘𝐴)))    &   (𝜑𝐴 ⊆ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...(𝐺‘(#‘𝐴)))) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...(𝐺‘(#‘𝐴))) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘(𝐺𝑁)) = (seq1( + , 𝐻)‘𝑁))

Theoremseqcoll2 13232* The function 𝐹 contains a sparse set of nonzero values to be summed. The function 𝐺 is an order isomorphism from the set of nonzero values of 𝐹 to a 1-based finite sequence, and 𝐻 collects these nonzero values together. Under these conditions, the sum over the values in 𝐻 yields the same result as the sum over the original set 𝐹. (Contributed by Mario Carneiro, 13-Dec-2014.)
((𝜑𝑘𝑆) → (𝑍 + 𝑘) = 𝑘)    &   ((𝜑𝑘𝑆) → (𝑘 + 𝑍) = 𝑘)    &   ((𝜑 ∧ (𝑘𝑆𝑛𝑆)) → (𝑘 + 𝑛) ∈ 𝑆)    &   (𝜑𝑍𝑆)    &   (𝜑𝐺 Isom < , < ((1...(#‘𝐴)), 𝐴))    &   (𝜑𝐴 ≠ ∅)    &   (𝜑𝐴 ⊆ (𝑀...𝑁))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑆)    &   ((𝜑𝑘 ∈ ((𝑀...𝑁) ∖ 𝐴)) → (𝐹𝑘) = 𝑍)    &   ((𝜑𝑛 ∈ (1...(#‘𝐴))) → (𝐻𝑛) = (𝐹‘(𝐺𝑛)))       (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(#‘𝐴)))

5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)

Theoremhashprlei 13233 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵} ∈ Fin ∧ (#‘{𝐴, 𝐵}) ≤ 2)

Theoremhash2pr 13234* A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏 𝑉 = {𝑎, 𝑏})

Theoremhash2prde 13235* A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉𝑊 ∧ (#‘𝑉) = 2) → ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏}))

Theoremhash2exprb 13236* A set of size two is an unordered pair if and only if it contains two different elements. (Contributed by Alexander van der Vekens, 14-Jan-2018.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑏(𝑎𝑏𝑉 = {𝑎, 𝑏})))

Theoremhash2prb 13237* A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑉𝑊 → ((#‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))

Theoremprprrab 13238 The set of proper pairs of elements of a given set expressed in two ways. (Contributed by AV, 24-Nov-2020.)
{𝑥 ∈ (𝒫 𝐴 ∖ {∅}) ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 2}

Theoremnehash2 13239 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)
(𝜑𝑃𝑉)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → 2 ≤ (#‘𝑃))

Theoremhash2prd 13240 A set of size two is an unordered pair if it contains two different elements. (Contributed by Alexander van der Vekens, 9-Dec-2018.) (Proof shortened by AV, 1-Nov-2020.)
((𝑃𝑉 ∧ (#‘𝑃) = 2) → ((𝑋𝑃𝑌𝑃𝑋𝑌) → 𝑃 = {𝑋, 𝑌}))

Theoremhash2pwpr 13241 If the size of a subset of an unordered pair is 2, the subset is the pair itself. (Contributed by Alexander van der Vekens, 9-Dec-2018.)
(((#‘𝑃) = 2 ∧ 𝑃 ∈ 𝒫 {𝑋, 𝑌}) → 𝑃 = {𝑋, 𝑌})

Theoremhashle2pr 13242* A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.)
((𝑃𝑉𝑃 ≠ ∅) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑏 𝑃 = {𝑎, 𝑏}))

Theoremhashle2prv 13243* A nonempty subset of a powerset of a class 𝑉 has size less than or equal to two iff it is an unordered pair of elements of 𝑉. (Contributed by AV, 24-Nov-2021.)
(𝑃 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝑃) ≤ 2 ↔ ∃𝑎𝑉𝑏𝑉 𝑃 = {𝑎, 𝑏}))

Theorempr2pwpr 13244* The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)
((𝐴𝑉𝐵𝑊𝐴𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2𝑜} = {{𝐴, 𝐵}})

Theoremhashge2el2dif 13245* A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)
((𝐷𝑉 ∧ 2 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷 𝑥𝑦)

Theoremhashge2el2difr 13246* A set with at least 2 different elements has size at least 2. (Contributed by AV, 14-Oct-2020.)
((𝐷𝑉 ∧ ∃𝑥𝐷𝑦𝐷 𝑥𝑦) → 2 ≤ (#‘𝐷))

Theoremhashge2el2difb 13247* A set has size at least 2 iff it has at least 2 different elements. (Contributed by AV, 14-Oct-2020.)
(𝐷𝑉 → (2 ≤ (#‘𝐷) ↔ ∃𝑥𝐷𝑦𝐷 𝑥𝑦))

Theoremhashdmpropge2 13248 The size of the domain of a class which contains two ordered pairs with different first componens is greater than or mequal to 2. (Contributed by AV, 12-Nov-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐷𝑌)    &   (𝜑𝐹𝑍)    &   (𝜑𝐴𝐵)    &   (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹)       (𝜑 → 2 ≤ (#‘dom 𝐹))

Theoremhashtplei 13249 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵, 𝐶} ∈ Fin ∧ (#‘{𝐴, 𝐵, 𝐶}) ≤ 3)

Theoremhashtpg 13250 The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 18-Sep-2021.)
((𝐴𝑈𝐵𝑉𝐶𝑊) → ((𝐴𝐵𝐵𝐶𝐶𝐴) ↔ (#‘{𝐴, 𝐵, 𝐶}) = 3))

Theoremhashge3el3dif 13251* A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 13245, which has an elementary proof, the dominance relation and 1-1 functions from a set with three elements which are known to be different are used to prove this theorem. Although there is also an elementary proof for this theorem, it might be much longer. After all, this proof should be kept because it can be used as template for proofs for higher cardinalities. (Contributed by AV, 20-Mar-2019.) (Proof modification is discouraged.)
((𝐷𝑉 ∧ 3 ≤ (#‘𝐷)) → ∃𝑥𝐷𝑦𝐷𝑧𝐷 (𝑥𝑦𝑥𝑧𝑦𝑧))

Theoremelss2prb 13252* An element of the set of subsets with two elements is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑃 ∈ {𝑧 ∈ 𝒫 𝑉 ∣ (#‘𝑧) = 2} ↔ ∃𝑥𝑉𝑦𝑉 (𝑥𝑦𝑃 = {𝑥, 𝑦}))

Theoremhash2sspr 13253* A subset of size two is an unordered pair of elements of its superset. (Contributed by Alexander van der Vekens, 12-Jul-2017.) (Proof shortened by AV, 4-Nov-2020.)
((𝑃 ∈ 𝒫 𝑉 ∧ (#‘𝑃) = 2) → ∃𝑎𝑉𝑏𝑉 𝑃 = {𝑎, 𝑏})

Theoremexprelprel 13254* If there is an element of the set of subsets with two elements in a set, an unordered pair of sets is in the set. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
(∃𝑝 ∈ {𝑒 ∈ 𝒫 𝑉 ∣ (#‘𝑒) = 2}𝑝𝑋 → ∃𝑣𝑉𝑤𝑉 {𝑣, 𝑤} ∈ 𝑋)

Theoremhash3tr 13255* A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑉𝑊 ∧ (#‘𝑉) = 3) → ∃𝑎𝑏𝑐 𝑉 = {𝑎, 𝑏, 𝑐})

Theoremhash1to3 13256* If the size of a set is between 1 and 3 (inclusively), the set is a singleton or an unordered pair or an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)
((𝑉 ∈ Fin ∧ 1 ≤ (#‘𝑉) ∧ (#‘𝑉) ≤ 3) → ∃𝑎𝑏𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))

5.6.11.2  Functions with a domain containing at least two different elements

Theoremfundmge2nop0 13257 A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 13258 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 15850. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Proof shortened by AV, 15-Nov-2021.)
((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Theoremfundmge2nop 13258 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.) (Proof shortened by AV, 9-Jun-2021.)
((Fun 𝐺 ∧ 2 ≤ (#‘dom 𝐺)) → ¬ 𝐺 ∈ (V × V))

Theoremfun2dmnop0 13259 A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 13260 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 15850. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun (𝐺 ∖ {∅}) ∧ 𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theoremfun2dmnop 13260 A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 9-Jun-2021.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐺𝐴𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

5.6.11.3  Finite induction on the size of the first component of a binary relation

Theorembrfi1indlem 13261 TODO-AV1: no lemma, but self-reliant theorem! Lemma for brfi1ind 13264: The size of a set is the size of this set with one element removed, increased by 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
((𝑉𝑊𝑁𝑉𝑌 ∈ ℕ0) → ((#‘𝑉) = (𝑌 + 1) → (#‘(𝑉 ∖ {𝑁})) = 𝑌))

Theoremfi1uzind 13262* Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 13267) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theorembrfi1uzind 13263* Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with 𝐿 = 0 (see brfi1ind 13264) or 𝐿 = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.)
Rel 𝐺    &   𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theorembrfi1ind 13264* Properties of a binary relation with a finite first component, proven by finite induction on the size of the first component. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Revised by AV, 28-Mar-2021.)
Rel 𝐺    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)

Theorembrfi1indALT 13265* Alternate proof of brfi1ind 13264, which does not use brfi1uzind 13263. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Rel 𝐺    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)

Theoremopfi1uzind 13266* Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 13267) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐸 ∈ V    &   𝐹 ∈ V    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theoremopfi1ind 13267* Properties of an ordered pair with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, e.g. fusgrfis 26203. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
𝐸 ∈ V    &   𝐹 ∈ V    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → 𝜑)

Theoremfi1uzindOLD 13268* Obsolete version of fi1uzind 13262 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theorembrfi1uzindOLD 13269* Obsolete version of brfi1uzind 13263 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theorembrfi1indOLD 13270* Obsolete version of brfi1ind 13264 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)

Theorembrfi1indALTOLD 13271* Obsolete version of brfi1indALT 13265 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Rel 𝐺    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((𝑉𝐺𝐸𝑉 ∈ Fin) → 𝜑)

Theoremopfi1uzindOLD 13272* Obsolete version of opfi1uzind 13266 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸𝑌    &   𝐹𝑈    &   𝐿 ∈ ℕ0    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 𝐿) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Theoremopfi1indOLD 13273* Obsolete version of opfi1ind 13267 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐸𝑌    &   𝐹𝑈    &   ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))    &   ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))    &   ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = 0) → 𝜓)    &   ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)       ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → 𝜑)

5.7  Words over a set

This section is about words (or strings) over a set (alphabet) defined as finite sequences of symbols (or characters) being elements of the alphabet. Although it is often required that the underlying set/alphabet be nonempty, finite and not a proper class, these restrictions are not made in the current definition df-word 13282, see, for example, s1cli 13367: ⟨“𝐴”⟩ ∈ Word V. Note that the empty word (i.e. the empty set) is the only word over an empty alphabet, see 0wrd0 13314. The set Word 𝑆 of words over 𝑆 is the free monoid over 𝑆, where the monoid law is concatenation and the monoid unit is the empty word. Besides the definition of words themselves, several operations on words are defined in this section:

NameReferenceExplanationExample Remarks
Length (or size) of a word df-hash 13101: (#‘𝑊) Operation which determines the number of the symbols within the word. 𝑊:(0..^𝑁)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (#‘𝑊) = 𝑁 This is not a special definition for words, but for arbitrary sets.
First symbol of a word df-fv 5884: (𝑊‘0) Operation which extracts the first symbol of a word. The result is the symbol at the first place in the sequence representing the word. 𝑊:(0..^1)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊‘0) ∈ 𝑆 This is not a special definition for words, but for arbitrary functions with a half-open range of nonnegative integers as domain.
Last symbol of a word df-lsw 13283: ( lastS ‘𝑊) Operation which extracts the last symbol of a word. The result is the symbol at the last place in the sequence representing the word. 𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ ( lastS ‘𝑊) = (𝑊‘2) Note that the index of the last symbol is less by 1 than the length of the word.
Subword (or substring) of a word df-substr 13286: (𝑊 substr ⟨𝐼, 𝐽⟩) Operation which extracts a portion of a word. The result is a consecutive, reindexed part of the sequence representing the word. 𝑊:(0..^3)⟶𝑆 → (𝑊 ∈ Word 𝑆 ∧ (𝑊 substr ⟨1, 2⟩) ∈ Word 𝑆 ∧ (#‘(𝑊 substr ⟨1, 2⟩)) = 1 Note that the last index of the range defining the subword is greater by 1 than the index of the last symbol of the subword in the sequence of the original word.
Concatenation of two words df-concat 13284: (𝑊 ++ 𝑈) Operation combining two words to one new word. The result is a combined, reindexed sequence build from the sequences representing the two words. (𝑊 ∈ Word 𝑆𝑈 ∈ Word 𝑆) → (#‘(𝑊 ++ 𝑈)) = ((#‘𝑊) + (#‘𝑈)) Note that the index of the first symbol of the second concatenated word is the length of the first word in the concatenation.
Reversal of a word df-reverse 13288: (reverse‘𝑊) Operation which reverses the order of symbols in a word. (𝑊 ∈ Word 𝑉 → (#‘(reverse‘𝑊)) = (#‘𝑊))
Cyclical shift of a word df-csh 13516: (𝑊 cyclShift 𝑁) Operation cyclically shifting the symbols by a number of positions. (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift (#‘𝑊)) = 𝑊)
Splicing of a word df-splice 13287: (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) Operation which replaces portions of words. ((𝑆 ∈ Word 𝐴𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Singleton word df-s1 13285: ⟨“𝑆”⟩ Constructor building a word of length 1 from a symbol. (#‘⟨“𝑆”⟩) = 1
Conventions:
• Words are usually represented by class variable 𝑊, if two words are involved by 𝑊 and 𝑈, or by 𝐴 and 𝐵.
• The alphabets are usually represented by class variable 𝑉 (because any arbitrary set can be an alphabet), sometimes also by 𝑆 (especially as codomain of the function representing a word, because the alphabet is the set of symbols).
• The symbols are usually represented by class variables 𝑆 or 𝐴, if two symbols are involved by 𝑆 and 𝑇, or by 𝐴 and 𝐵.
• The indices of the sequence representing a word are usually represented by class variable 𝐼, if two indices are involved (especially for subwords) by 𝐼 and 𝐽, or by 𝑀 and 𝑁.
• The length of a word is usually represented by class variables 𝑁 or 𝐿.
• The number of position to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.

5.7.1  Definitions and basic theorems

Syntaxcword 13274 Syntax for the Word operator.
class Word 𝑆

Syntaxclsw 13275 Extend class notation with the Last Symbol of a word.
class lastS

Syntaxcconcat 13276 Syntax for the concatenation operator.
class ++

Syntaxcs1 13277 Syntax for the singleton word constructor.
class ⟨“𝐴”⟩

Syntaxcsubstr 13278 Syntax for the subword operator.
class substr

Syntaxcsplice 13279 Syntax for the word splicing operator.
class splice

Syntaxcreverse 13280 Syntax for the word reverse operator.
class reverse

Syntaxcreps 13281 Extend class notation with words consisting of one repeated symbol.
class repeatS

Definitiondf-word 13282* Define the class of words over a set. A word (sometimes also called a string) is a finite sequence of symbols from a set (alphabet) 𝑆. Definition in section 9.1 of [AhoHopUll] p. 318. The domain is forced to be an initial segment of 0 so that two words with the same symbols in the same order be equal. The set Word 𝑆 is sometimes denoted by S*, using the Kleene star, although the Kleene star, or Kleene closure, is sometimes reserved to denote an operation on languages. The set Word 𝑆 is the free monoid over 𝑆, where the monoid law is concatenation and the monoid unit is the empty word (see frmdval 17369). (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 14-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
Word 𝑆 = {𝑤 ∣ ∃𝑙 ∈ ℕ0 𝑤:(0..^𝑙)⟶𝑆}

Definitiondf-lsw 13283 Extract the last symbol of a word. May be not meaningful for other sets which are not words. The name lastS (as abbreviation of "lastSymbol") is a compromise between usually used names for corresponding functions in computer programs (as last() or lastChar()), the terminology used for words in set.mm ("symbol" instead of "character") and brevity ("lastS" is shorter than "lastChar" and "lastSymbol"). Labels of theorems about last symbols of a word will contain the abbreviation "lsw" (Last Symbol of a Word). (Contributed by Alexander van der Vekens, 18-Mar-2018.)
lastS = (𝑤 ∈ V ↦ (𝑤‘((#‘𝑤) − 1)))

Definitiondf-concat 13284* Define the concatenation operator which combines two words. Definition in section 9.1 of [AhoHopUll] p. 318. (Contributed by FL, 14-Jan-2014.) (Revised by Stefan O'Rear, 15-Aug-2015.)
++ = (𝑠 ∈ V, 𝑡 ∈ V ↦ (𝑥 ∈ (0..^((#‘𝑠) + (#‘𝑡))) ↦ if(𝑥 ∈ (0..^(#‘𝑠)), (𝑠𝑥), (𝑡‘(𝑥 − (#‘𝑠))))))

Definitiondf-s1 13285 Define the canonical injection from symbols to words. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
⟨“𝐴”⟩ = {⟨0, ( I ‘𝐴)⟩}

Definitiondf-substr 13286* Define an operation which extracts portions of words. Definition in section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.)
substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅))

Definitiondf-splice 13287* Define an operation which replaces portions of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)
splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))

Definitiondf-reverse 13288* Define an operation which reverses the order of symbols in a word. (Contributed by Stefan O'Rear, 26-Aug-2015.)
reverse = (𝑠 ∈ V ↦ (𝑥 ∈ (0..^(#‘𝑠)) ↦ (𝑠‘(((#‘𝑠) − 1) − 𝑥))))

Definitiondf-reps 13289* Definition to construct a word consisting of one repeated symbol, often called "repeated symbol word" for short in the following. (Contributed by Alexander van der Vekens, 4-Nov-2018.)
repeatS = (𝑠 ∈ V, 𝑛 ∈ ℕ0 ↦ (𝑥 ∈ (0..^𝑛) ↦ 𝑠))

Theoremiswrd 13290* Property of being a word over a set with a quantifier over the length. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)
(𝑊 ∈ Word 𝑆 ↔ ∃𝑙 ∈ ℕ0 𝑊:(0..^𝑙)⟶𝑆)

Theoremwrdval 13291* Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑆𝑉 → Word 𝑆 = 𝑙 ∈ ℕ0 (𝑆𝑚 (0..^𝑙)))

Theoremiswrdi 13292 A zero-based sequence is a word. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)
(𝑊:(0..^𝐿)⟶𝑆𝑊 ∈ Word 𝑆)

Theoremwrdf 13293 A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)
(𝑊 ∈ Word 𝑆𝑊:(0..^(#‘𝑊))⟶𝑆)

Theoremiswrdb 13294 A word over an alphabet is a function of an open range of nonnegative integers (of length equal to the length of the word) into the alphabet. (Contributed by Alexander van der Vekens, 30-Jul-2018.)
(𝑊 ∈ Word 𝑆𝑊:(0..^(#‘𝑊))⟶𝑆)

Theoremwrddm 13295 The indices of a word (i.e. its domain regarded as function) are elements of an open range of nonnegative integers (of length equal to the length of the word). (Contributed by AV, 2-May-2020.)
(𝑊 ∈ Word 𝑆 → dom 𝑊 = (0..^(#‘𝑊)))

Theoremsswrd 13296 The set of words respects ordering on the base set. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 13-May-2020.)
(𝑆𝑇 → Word 𝑆 ⊆ Word 𝑇)

Theoremsnopiswrd 13297 A singleton of an ordered pair (with 0 as first component) is a word. (Contributed by AV, 23-Nov-2018.) (Proof shortened by AV, 18-Apr-2021.)
(𝑆𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)

Theoremwrdexg 13298 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆𝑉 → Word 𝑆 ∈ V)

Theoremwrdexb 13299 The set of words over a set is a set, bidirectional version. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by AV, 23-Nov-2018.)
(𝑆 ∈ V ↔ Word 𝑆 ∈ V)

Theoremwrdexi 13300 The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)
𝑆 ∈ V       Word 𝑆 ∈ V

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