P. 146 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

phphashd 14501

Corollary of the Pigeonhole Principle using equality. Equivalent of phpeqd 9249 expressed using the hash function. (Contributed by Rohan Ridenour, 3-Aug-2023.)

⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)

Theorem

phphashrd 14502

Corollary of the Pigeonhole Principle using equality. Equivalent of phphashd 14501 with reversed arguments. (Contributed by Rohan Ridenour, 3-Aug-2023.)

⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)

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

Theorem

hashprlei 14503

An unordered pair has at most two elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

⊢ ({𝐴, 𝐵} ∈ Fin ∧ (♯‘{𝐴, 𝐵}) ≤ 2)

Theorem

hash2pr 14504*

A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏})

Theorem

hash2prde 14505*

A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))

Theorem

hash2exprb 14506*

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.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 2 ↔ ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})))

Theorem

hash2prb 14507*

A set of size two is a proper unordered pair. (Contributed by AV, 1-Nov-2020.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 2 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})))

Theorem

prprrab 14508

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}

Theorem

nehash2 14509

The cardinality of a set with two distinct elements. (Contributed by Thierry Arnoux, 27-Aug-2019.)

⊢ (𝜑 → 𝑃 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑃)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 2 ≤ (♯‘𝑃))

Theorem

hash2prd 14510

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.)

⊢ ((𝑃 ∈ 𝑉 ∧ (♯‘𝑃) = 2) → ((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) → 𝑃 = {𝑋, 𝑌}))

Theorem

hash2pwpr 14511

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})

Theorem

hashle2pr 14512*

A nonempty set of size less than or equal to two is an unordered pair of sets. (Contributed by AV, 24-Nov-2021.)

⊢ ((𝑃 ∈ 𝑉 ∧ 𝑃 ≠ ∅) → ((♯‘𝑃) ≤ 2 ↔ ∃𝑎∃𝑏 𝑃 = {𝑎, 𝑏}))

Theorem

hashle2prv 14513*

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 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑃 = {𝑎, 𝑏}))

Theorem

pr2pwpr 14514*

The set of subsets of a pair having length 2 is the set of the pair as singleton. (Contributed by AV, 9-Dec-2018.)

⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → {𝑝 ∈ 𝒫 {𝐴, 𝐵} ∣ 𝑝 ≈ 2_o} = {{𝐴, 𝐵}})

Theorem

hashge2el2dif 14515*

A set with size at least 2 has at least 2 different elements. (Contributed by AV, 18-Mar-2019.)

⊢ ((𝐷 ∈ 𝑉 ∧ 2 ≤ (♯‘𝐷)) → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦)

Theorem

hashge2el2difr 14516*

A set with at least 2 different elements has size at least 2. (Contributed by AV, 14-Oct-2020.)

⊢ ((𝐷 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦) → 2 ≤ (♯‘𝐷))

Theorem

hashge2el2difb 14517*

A set has size at least 2 iff it has at least 2 different elements. (Contributed by AV, 14-Oct-2020.)

⊢ (𝐷 ∈ 𝑉 → (2 ≤ (♯‘𝐷) ↔ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 𝑥 ≠ 𝑦))

Theorem

hashdmpropge2 14518

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.)

⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑍)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ⊆ 𝐹)    ⇒   ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐹))

Theorem

hashtplei 14519

An unordered triple has at most three elements. (Contributed by Mario Carneiro, 11-Feb-2015.)

⊢ ({𝐴, 𝐵, 𝐶} ∈ Fin ∧ (♯‘{𝐴, 𝐵, 𝐶}) ≤ 3)

Theorem

hashtpg 14520

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.)

⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝐴) ↔ (♯‘{𝐴, 𝐵, 𝐶}) = 3))

Theorem

hash7g 14521

The size of an unordered set of seven different elements. (Contributed by AV, 2-Aug-2025.)

⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐷 ∈ 𝑉 ∧ (𝐸 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) ∧ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐷) ∧ (𝐴 ≠ 𝐸 ∧ 𝐴 ≠ 𝐹 ∧ 𝐴 ≠ 𝐺)) ∧ ((𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) ∧ (𝐵 ≠ 𝐸 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐺)) ∧ (𝐶 ≠ 𝐷 ∧ (𝐶 ≠ 𝐸 ∧ 𝐶 ≠ 𝐹 ∧ 𝐶 ≠ 𝐺))) ∧ ((𝐷 ≠ 𝐸 ∧ 𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐺) ∧ (𝐸 ≠ 𝐹 ∧ 𝐸 ≠ 𝐺 ∧ 𝐹 ≠ 𝐺)))) → (♯‘(({𝐴, 𝐵, 𝐶} ∪ {𝐷}) ∪ {𝐸, 𝐹, 𝐺})) = 7)

Theorem

hashge3el3dif 14522*

A set with size at least 3 has at least 3 different elements. In contrast to hashge2el2dif 14515, 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.)

⊢ ((𝐷 ∈ 𝑉 ∧ 3 ≤ (♯‘𝐷)) → ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 ∃𝑧 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧))

Theorem

elss2prb 14523*

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} ↔ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝑥 ≠ 𝑦 ∧ 𝑃 = {𝑥, 𝑦}))

Theorem

hash2sspr 14524*

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) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑃 = {𝑎, 𝑏})

Theorem

exprelprel 14525*

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}𝑝 ∈ 𝑋 → ∃𝑣 ∈ 𝑉 ∃𝑤 ∈ 𝑉 {𝑣, 𝑤} ∈ 𝑋)

Theorem

hash3tr 14526*

A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐})

Theorem

hash1to3 14527*

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) → ∃𝑎∃𝑏∃𝑐(𝑉 = {𝑎} ∨ 𝑉 = {𝑎, 𝑏} ∨ 𝑉 = {𝑎, 𝑏, 𝑐}))

Theorem

hash3tpde 14528*

A set of size three is an unordered triple of three different elements. (Contributed by AV, 21-Jul-2025.)

⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐}))

Theorem

hash3tpexb 14529*

A set of size three is an unordered triple if and only if it contains three different elements. (Contributed by AV, 21-Jul-2025.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎∃𝑏∃𝑐((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))

Theorem

hash3tpb 14530*

A set of size three is a proper unordered triple. (Contributed by AV, 21-Jul-2025.)

⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ((𝑎 ≠ 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ∧ 𝑉 = {𝑎, 𝑏, 𝑐})))

Theorem

tpf1ofv0 14531*

The value of a one-to-one function onto a triple at 0. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹‘0) = 𝐴)

Theorem

tpf1ofv1 14532*

The value of a one-to-one function onto a triple at 1. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐵 ∈ 𝑉 → (𝐹‘1) = 𝐵)

Theorem

tpf1ofv2 14533*

The value of a one-to-one function onto a triple at 2. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    ⇒   ⊢ (𝐶 ∈ 𝑉 → (𝐹‘2) = 𝐶)

Theorem

tpf 14534*

A function into a (proper) triple. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)⟶𝑇)

Theorem

tpfo 14535*

A function onto a (proper) triple. (Contributed by AV, 20-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝐹:(0..^3)–onto→𝑇)

Theorem

tpf1o 14536*

A bijection onto a (proper) triple. (Contributed by AV, 21-Jul-2025.)

⊢ 𝐹 = (𝑥 ∈ (0..^3) ↦ if(𝑥 = 0, 𝐴, if(𝑥 = 1, 𝐵, 𝐶)))    &   ⊢ 𝑇 = {𝐴, 𝐵, 𝐶}    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ (♯‘𝑇) = 3) → 𝐹:(0..^3)–1-1-onto→𝑇)

5.6.11.2  Functions with a domain containing at least two different elements

Theorem

fundmge2nop0 14537

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fundmge2nop 14538 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17184. (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))

Theorem

fundmge2nop 14538

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))

Theorem

fun2dmnop0 14539

A function with a domain containing (at least) two different elements is not an ordered pair. This stronger version of fun2dmnop 14540 (with the less restrictive requirement that (𝐺 ∖ {∅}) needs to be a function instead of 𝐺) is useful for proofs for extensible structures, see structn0fun 17184. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.)

⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

Theorem

fun2dmnop 14540

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 𝐺 ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))

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

Theorem

hashdifsnp1 14541

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.)

⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉 ∧ 𝑌 ∈ ℕ₀) → ((♯‘𝑉) = (𝑌 + 1) → (♯‘(𝑉 ∖ {𝑁})) = 𝑌))

Theorem

fi1uzind 14542*

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 ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14547) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐹 ∈ V    &   ⊢ 𝐿 ∈ ℕ₀    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

brfi1uzind 14543*

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 14544) 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 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

brfi1ind 14544*

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 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Theorem

brfi1indALT 14545*

Alternate proof of brfi1ind 14544, which does not use brfi1uzind 14543. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

⊢ Rel 𝐺    &   ⊢ 𝐹 ∈ V    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Theorem

opfi1uzind 14546*

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 ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14547) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ 𝐿 ∈ ℕ₀    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Theorem

opfi1ind 14547*

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 ordered pairs of vertices and edges) with a finite number of vertices, e.g., fusgrfis 29361. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    &   ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))    &   ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)    &   ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))    &   ⊢ ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 0) → 𝜓)    &   ⊢ ((((𝑦 + 1) ∈ ℕ₀ ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ ∈ 𝐺 ∧ 𝑉 ∈ 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 14549, see, for example, s1cli 14639: ⟨“𝐴”⟩ ∈ Word V. Note that the empty word ∅ (i.e., the empty set) is the only word over an empty alphabet, see 0wrd0 14574. 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:

Name	Reference	Explanation	Example	Remarks
Length (or size) of a word	df-hash 14366: (♯‘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 6570: (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 14597: (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 14675: (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 14605: (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 𝑆 ∧ 𝑈 ∈ 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 14793: (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 14823: (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 14784: (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩)	Operation which replaces portions of words.	((𝑆 ∈ Word 𝐴 ∧ 𝑅 ∈ Word 𝐴) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) ∈ Word 𝐴)
Singleton word	df-s1 14630: ⟨“𝑆”⟩	Constructor building a word of length 1 from a symbol.	(♯‘⟨“𝑆”⟩) = 1

Conventions:

• Words are usually represented by class variable 𝑊, or 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, or 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 positions by which to cyclically shift a word is usually represented by class variables 𝑁 or 𝐿.

5.7.1  Definitions and basic theorems

Syntax

cword 14548

Syntax for the Word operator.

class Word 𝑆

Definition

df-word 14549*

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 ℕ₀ 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 equipped with concatenation is the free monoid over 𝑆, and the monoid unit is the empty word (see frmdval 18876). (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}

Theorem

iswrd 14550*

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)

Theorem

wrdval 14551*

Value of the set of words over a set. (Contributed by Stefan O'Rear, 10-Aug-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 = ∪ 𝑙 ∈ ℕ₀ (𝑆 ↑_m (0..^𝑙)))

Theorem

iswrdi 14552

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)⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrdf 14553

A word is a zero-based sequence with a recoverable upper limit. (Contributed by Stefan O'Rear, 15-Aug-2015.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

iswrdb 14554

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 𝑆 ↔ 𝑊:(0..^(♯‘𝑊))⟶𝑆)

Theorem

wrddm 14555

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)))

Theorem

sswrd 14556

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.)

⊢ (𝑆 ⊆ 𝑇 → Word 𝑆 ⊆ Word 𝑇)

Theorem

snopiswrd 14557

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.)

⊢ (𝑆 ∈ 𝑉 → {⟨0, 𝑆⟩} ∈ Word 𝑉)

Theorem

wrdexg 14558

The set of words over a set is a set. (Contributed by Mario Carneiro, 26-Feb-2016.) (Proof shortened by JJ, 18-Nov-2022.)

⊢ (𝑆 ∈ 𝑉 → Word 𝑆 ∈ V)

Theorem

wrdexb 14559

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)

Theorem

wrdexi 14560

The set of words over a set is a set, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 ∈ V    ⇒   ⊢ Word 𝑆 ∈ V

Theorem

wrdsymbcl 14561

A symbol within a word over an alphabet belongs to the alphabet. (Contributed by Alexander van der Vekens, 28-Jun-2018.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝐼) ∈ 𝑉)

Theorem

wrdfn 14562

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 𝑆 → 𝑊 Fn (0..^(♯‘𝑊)))

Theorem

wrdv 14563

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 𝑉 → 𝑊 ∈ Word V)

Theorem

wrdlndm 14564

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)

Theorem

iswrdsymb 14565*

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))(𝑊‘𝑖) ∈ 𝑉) → 𝑊 ∈ Word 𝑉)

Theorem

wrdfin 14566

A word is a finite set. (Contributed by Stefan O'Rear, 2-Nov-2015.) (Proof shortened by AV, 18-Nov-2018.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊 ∈ Fin)

Theorem

lencl 14567

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)

Theorem

lennncl 14568

The length of a nonempty word is a positive integer. (Contributed by Mario Carneiro, 1-Oct-2015.)

⊢ ((𝑊 ∈ Word 𝑆 ∧ 𝑊 ≠ ∅) → (♯‘𝑊) ∈ ℕ)

Theorem

wrdffz 14569

A word is a function from a finite interval of integers. (Contributed by AV, 10-Feb-2021.)

⊢ (𝑊 ∈ Word 𝑆 → 𝑊:(0...((♯‘𝑊) − 1))⟶𝑆)

Theorem

wrdeq 14570

Equality theorem for the set of words. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ (𝑆 = 𝑇 → Word 𝑆 = Word 𝑇)

Theorem

wrdeqi 14571

Equality theorem for the set of words, inference form. (Contributed by AV, 23-May-2021.)

⊢ 𝑆 = 𝑇    ⇒   ⊢ Word 𝑆 = Word 𝑇

Theorem

iswrddm0 14572

A function with empty domain is a word. (Contributed by AV, 13-Oct-2018.)

⊢ (𝑊:∅⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrd0 14573

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

Theorem

0wrd0 14574

The empty word is the only word over an empty alphabet. (Contributed by AV, 25-Oct-2018.)

⊢ (𝑊 ∈ Word ∅ ↔ 𝑊 = ∅)

Theorem

ffz0iswrd 14575

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)⟶𝑆 → 𝑊 ∈ Word 𝑆)

Theorem

wrdsymb 14576

A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.)

⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆))))

Theorem

nfwrd 14577

Hypothesis builder for Word 𝑆. (Contributed by Mario Carneiro, 26-Feb-2016.)

⊢ Ⅎ𝑥𝑆    ⇒   ⊢ Ⅎ𝑥Word 𝑆

Theorem

csbwrdg 14578*

Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ (𝑆 ∈ 𝑉 → ⦋𝑆 / 𝑥⦌Word 𝑥 = Word 𝑆)

Theorem

wrdnval 14579*

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.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {𝑤 ∈ Word 𝑉 ∣ (♯‘𝑤) = 𝑁} = (𝑉 ↑_m (0..^𝑁)))

Theorem

wrdmap 14580

Words as a mapping. (Contributed by Thierry Arnoux, 4-Mar-2020.)

⊢ ((𝑉 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ↔ 𝑊 ∈ (𝑉 ↑_m (0..^𝑁))))

Theorem

hashwrdn 14581*

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))

Theorem

wrdnfi 14582*

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)

Theorem

wrdsymb0 14583

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 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝑊) ≤ 𝐼) → (𝑊‘𝐼) = ∅))

Theorem

wrdlenge1n0 14584

A word with length at least 1 is not empty. (Contributed by AV, 14-Oct-2018.)

⊢ (𝑊 ∈ Word 𝑉 → (𝑊 ≠ ∅ ↔ 1 ≤ (♯‘𝑊)))

Theorem

len0nnbi 14585

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))

Theorem

wrdlenge2n0 14586

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 ≠ ∅)

Theorem

wrdsymb1 14587

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)

Theorem

wrdlen1 14588*

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) → ∃𝑣 ∈ 𝑉 (𝑊‘0) = 𝑣)

Theorem

fstwrdne 14589

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 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊‘0) ∈ 𝑉)

Theorem

fstwrdne0 14590

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)

Theorem

eqwrd 14591*

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 𝑆 ∧ 𝑊 ∈ Word 𝑇) → (𝑈 = 𝑊 ↔ ((♯‘𝑈) = (♯‘𝑊) ∧ ∀𝑖 ∈ (0..^(♯‘𝑈))(𝑈‘𝑖) = (𝑊‘𝑖))))

Theorem

elovmpowrd 14592*

Implications for the value of an operation defined by the maps-to notation with a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑})    ⇒   ⊢ (𝑍 ∈ (𝑉𝑂𝑌) → (𝑉 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ Word 𝑉))

Theorem

elovmptnn0wrd 14593*

Implications for the value of an operation defined by the maps-to notation with a function of nonnegative integers into a class abstraction of words as a result having an element. Note that 𝜑 may depend on 𝑧 as well as on 𝑣 and 𝑦 and 𝑛. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)

⊢ 𝑂 = (𝑣 ∈ V, 𝑦 ∈ V ↦ (𝑛 ∈ ℕ₀ ↦ {𝑧 ∈ Word 𝑣 ∣ 𝜑}))    ⇒   ⊢ (𝑍 ∈ ((𝑉𝑂𝑌)‘𝑁) → ((𝑉 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑍 ∈ Word 𝑉)))

Theorem

wrdred1 14594

A word truncated by a symbol is a word. (Contributed by AV, 29-Jan-2021.)

⊢ (𝐹 ∈ Word 𝑆 → (𝐹 ↾ (0..^((♯‘𝐹) − 1))) ∈ Word 𝑆)

Theorem

wrdred1hash 14595

The length of a word truncated by a symbol. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)

⊢ ((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (♯‘𝐹)) → (♯‘(𝐹 ↾ (0..^((♯‘𝐹) − 1)))) = ((♯‘𝐹) − 1))

5.7.2  Last symbol of a word

Syntax

clsw 14596

Extend class notation with the Last Symbol of a word.

class lastS

Definition

df-lsw 14597

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)))

Theorem

lsw 14598

Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.)

⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))

Theorem

lsw0 14599

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 19-Mar-2018.) (Proof shortened by AV, 2-May-2020.)

⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 0) → (lastS‘𝑊) = ∅)

Theorem

lsw0g 14600

The last symbol of an empty word does not exist. (Contributed by Alexander van der Vekens, 11-Nov-2018.)

⊢ (lastS‘∅) = ∅