Metamath Proof Explorer


Theorem wlkpwrd

Description: The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021)

Ref Expression
Hypothesis wlkp.v 𝑉 = ( Vtx ‘ 𝐺 )
Assertion wlkpwrd ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃𝑃 ∈ Word 𝑉 )

Proof

Step Hyp Ref Expression
1 wlkp.v 𝑉 = ( Vtx ‘ 𝐺 )
2 1 wlkp ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 )
3 ffz0iswrd ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉𝑃 ∈ Word 𝑉 )
4 2 3 syl ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃𝑃 ∈ Word 𝑉 )