Theorem iswwlksn

Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018) (Revised by AV, 8-Apr-2021)

	Ref	Expression
Assertion	iswwlksn	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		wwlksn	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } )
2	1	eleq2d	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ 𝑊 ∈ { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ) )
3		fveqeq2	⊢ ( 𝑤 = 𝑊 → ( ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ↔ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) )
4	3	elrab	⊢ ( 𝑊 ∈ { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) )
5	2 4	bitrdi	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑊 ∈ ( 𝑁 WWalksN 𝐺 ) ↔ ( 𝑊 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑊 ) = ( 𝑁 + 1 ) ) ) )