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Theorem List for Metamath Proof Explorer - 13801-13900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremffz0hash 13801 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 13802 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 13803 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 13804 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 13805 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 13806* Lemma for hashbc 13807: 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 13807* 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 13808* 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 13809* Lemma for hashf1 13811. (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 13810* Lemma for hashf1 13811. (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 13811* 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 13812* 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 13813 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 13814 Version of isorel 7058 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 13815* Lemma for fz1iso 13816. (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 13816* Any finite ordered set has an order isomorphism 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 13817* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 8715. (Contributed by Thierry Arnoux, 5-Jul-2017.)
𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑥 ∈ 𝒫 𝐴(♯‘𝑥) = 𝑛)
 
Theoremseqcoll 13818* 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 13819* 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)))
 
Theoremphphashd 13820 Corollary of the Pigeonhole Principle using equality. Equivalent of phpeqd 8690 expressed using the hash function. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑 → (♯‘𝐴) = (♯‘𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremphphashrd 13821 Corollary of the Pigeonhole Principle using equality. Equivalent of phphashd 13820 with reversed arguments. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐵 ∈ Fin)    &   (𝜑𝐴𝐵)    &   (𝜑 → (♯‘𝐴) = (♯‘𝐵))       (𝜑𝐴 = 𝐵)
 
5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)
 
Theoremhashprlei 13822 An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)
 
Theoremhash2pr 13823* 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 13824* 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 13825* 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 13826* A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)
(𝑉𝑊 → ((♯‘𝑉) = 2 ↔ ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝑉 = {𝑎, 𝑏})))
 
Theoremprprrab 13827 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 13828 The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)
(𝜑𝑃𝑉)    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐴𝐵)       (𝜑 → 2 ≤ (♯‘𝑃))
 
Theoremhash2prd 13829 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, 16-Jun-2022.)
((𝑃𝑉 ∧ (♯‘𝑃) = 2) → ((𝑋𝑃𝑌𝑃𝑋𝑌) → 𝑃 = {𝑋, 𝑌}))
 
Theoremhash2pwpr 13830 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 13831* 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 13832* 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 13833* 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 13834* 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 13835* 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 13836* 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 13837 The size of the domain of a class which contains two ordered pairs with different first components is greater than or equal 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 13838 An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)
({𝐴, 𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐵, 𝐶}) ≤ 3)
 
Theoremhashtpg 13839 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 13840* A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 13834, 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 13841* 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 13842* 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 13843* 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 13844* 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 13845* 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 13846 A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 13847 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 16487. (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 13847 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 13848 A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 13849 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 16487. (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 13849 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
 
Theoremhashdifsnp1 13850 If the size of a set is a nonnegative integer increased by 1, the size of the set with one of its elements removed is this nonnegative integer. (Contributed by Alexander van der Vekens, 7-Jan-2018.)
((𝑉𝑊𝑁𝑉𝑌 ∈ ℕ0) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌))
 
Theoremfi1uzind 13851* 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 13856) 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 13852* 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 13853) 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 13853* 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 13854* Alternate proof of brfi1ind 13853, which does not use brfi1uzind 13852. (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 13855* 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 13856) 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 13856* 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 27120. (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)
 
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 13858, see, for example, s1cli 13950: ⟨“𝐴”⟩ ∈ Word V. Note that the empty word (i.e. the empty set) is the only word over an empty alphabet, see 0wrd0 13883. 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 13687: (♯‘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 6332: (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 13906: (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 13994: (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 13914: (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 14112: (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 14142: (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 14103: (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 13941: ⟨“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 13857 Syntax for the Word operator.
class Word 𝑆
 
Definitiondf-word 13858* 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 18008). (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}
 
Theoremiswrd 13859* Property of being a word over a set with an existential 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 13860* 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 13861 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 13862 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 13863 A word over an alphabet is a function from 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 13864 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 13865 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 13866 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 13867 The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)
(𝑆𝑉 → Word 𝑆 ∈ V)
 
Theoremwrdexb 13868 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 13869 The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)
𝑆 ∈ V       Word 𝑆 ∈ V
 
Theoremwrdsymbcl 13870 A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
((𝑊 ∈ Word 𝑉𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊𝐼) ∈ 𝑉)
 
Theoremwrdfn 13871 A word is a function with a zero-based sequence of integers as domain. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
(𝑊 ∈ Word 𝑆𝑊 Fn (0..^(♯‘𝑊)))
 
Theoremwrdv 13872 A word over an alphabet is a word over the universal class. (Contributed by AV, 8-Feb-2021.) (Proof shortened by JJ, 18-Nov-2022.)
(𝑊 ∈ Word 𝑉𝑊 ∈ Word V)
 
Theoremwrdlndm 13873 The length of a word is not in the domain of the word (regarded as a function). (Contributed by AV, 3-Mar-2021.) (Proof shortened by JJ, 18-Nov-2022.)
(𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∉ dom 𝑊)
 
Theoremiswrdsymb 13874* An arbitrary word is a word over an alphabet if all of its symbols belong to the alphabet. (Contributed by AV, 23-Jan-2021.)
((𝑊 ∈ Word V ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ∈ 𝑉) → 𝑊 ∈ Word 𝑉)
 
Theoremwrdfin 13875 A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)
(𝑊 ∈ Word 𝑆𝑊 ∈ Fin)
 
Theoremlencl 13876 The length of a word is a nonnegative integer. This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 27-Aug-2015.)
(𝑊 ∈ Word 𝑆 → (♯‘𝑊) ∈ ℕ0)
 
Theoremlennncl 13877 The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)
((𝑊 ∈ Word 𝑆𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)
 
Theoremwrdffz 13878 A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)
(𝑊 ∈ Word 𝑆𝑊:(0...((♯‘𝑊) − 1))⟶𝑆)
 
Theoremwrdeq 13879 Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)
(𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)
 
Theoremwrdeqi 13880 Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)
𝑆 = 𝑇       Word 𝑆 = Word 𝑇
 
Theoremiswrddm0 13881 A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)
(𝑊:∅⟶𝑆𝑊 ∈ Word 𝑆)
 
Theoremwrd0 13882 The empty set is a word (the empty word, frequently denoted ε in this context). This corresponds to the definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Proof shortened by AV, 13-May-2020.)
∅ ∈ Word 𝑆
 
Theorem0wrd0 13883 The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)
(𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)
 
Theoremffz0iswrd 13884 A sequence with zero-based indices is a word. (Contributed by AV, 31-Jan-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by JJ, 18-Nov-2022.)
(𝑊:(0...𝐿)⟶𝑆𝑊 ∈ Word 𝑆)
 
Theoremwrdsymb 13885 A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)
(𝑆 ∈ Word 𝐴𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆))))
 
Theoremnfwrd 13886 Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)
𝑥𝑆       𝑥Word 𝑆
 
Theoremcsbwrdg 13887* Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑆𝑉𝑆 / 𝑥Word 𝑥 = Word 𝑆)
 
Theoremwrdnval 13888* Words of a fixed length are mappings from a fixed half-open integer interval. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Proof shortened by AV, 13-May-2020.)
((𝑉𝑋𝑁 ∈ ℕ0) → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = (𝑉m (0..^𝑁)))
 
Theoremwrdmap 13889 Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)
((𝑉𝑋𝑁 ∈ ℕ0) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉m (0..^𝑁))))
 
Theoremhashwrdn 13890* If there is only a finite number of symbols, the number of words of a fixed length over these sysmbols is the number of these symbols raised to the power of the length. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁}) = ((♯‘𝑉)↑𝑁))
 
Theoremwrdnfi 13891* If there is only a finite number of symbols, the number of words of a fixed length over these symbols is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) Remove unnecessary antecedent. (Revised by JJ, 18-Nov-2022.)
(𝑉 ∈ Fin → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} ∈ Fin)
 
Theoremwrdsymb0 13892 A symbol at a position "outside" of a word. (Contributed by Alexander van der Vekens, 26-May-2018.) (Proof shortened by AV, 2-May-2020.)
((𝑊 ∈ Word 𝑉𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝑊) ≤ 𝐼) → (𝑊𝐼) = ∅))
 
Theoremwrdlenge1n0 13893 A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)
(𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊)))
 
Theoremlen0nnbi 13894 The length of a word is a positive integer iff the word is not empty. (Contributed by AV, 22-Mar-2022.)
(𝑊 ∈ Word 𝑆 → (𝑊 ≠ ∅ ↔ (♯‘𝑊) ∈ ℕ))
 
Theoremwrdlenge2n0 13895 A word with length at least 2 is not empty. (Contributed by AV, 18-Jun-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑊)) → 𝑊 ≠ ∅)
 
Theoremwrdsymb1 13896 The first symbol of a nonempty word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑊)) → (𝑊‘0) ∈ 𝑉)
 
Theoremwrdlen1 13897* A word of length 1 starts with a symbol. (Contributed by AV, 20-Jul-2018.) (Proof shortened by AV, 19-Oct-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 1) → ∃𝑣𝑉 (𝑊‘0) = 𝑣)
 
Theoremfstwrdne 13898 The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)
 
Theoremfstwrdne0 13899 The first symbol of a nonempty word is element of the alphabet for the word. (Contributed by AV, 29-Sep-2018.) (Proof shortened by AV, 14-Oct-2018.)
((𝑁 ∈ ℕ ∧ (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁)) → (𝑊‘0) ∈ 𝑉)
 
Theoremeqwrd 13900* Two words are equal iff they have the same length and the same symbol at each position. (Contributed by AV, 13-Apr-2018.) (Revised by JJ, 30-Dec-2023.)
((𝑈 ∈ Word 𝑆𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈𝑖) = (𝑊𝑖))))
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