Theorem wlkelwrd

Description: The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 2-Jan-2021)

	Ref	Expression
Hypotheses	wlkcomp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	wlkcomp.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	wlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
	wlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
Assertion	wlkelwrd	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		wlkcomp.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		wlkcomp.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		wlkcomp.1	⊢ 𝐹 = ( 1st ‘ 𝑊 )
4		wlkcomp.2	⊢ 𝑃 = ( 2nd ‘ 𝑊 )
5	1 2 3 4	wlkcompim	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )
6		3simpa	⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) )
7	5 6	syl	⊢ ( 𝑊 ∈ ( Walks ‘ 𝐺 ) → ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) )