Theorem wspthsswwlkn

Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021)

	Ref	Expression
Assertion	wspthsswwlkn	⊢ ( 𝑁 WSPathsN 𝐺 ) ⊆ ( 𝑁 WWalksN 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		wspthnp	⊢ ( 𝑤 ∈ ( 𝑁 WSPathsN 𝐺 ) → ( ( 𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V ) ∧ 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∧ ∃ 𝑓 𝑓 ( SPaths ‘ 𝐺 ) 𝑤 ) )
2	1	simp2d	⊢ ( 𝑤 ∈ ( 𝑁 WSPathsN 𝐺 ) → 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) )
3	2	ssriv	⊢ ( 𝑁 WSPathsN 𝐺 ) ⊆ ( 𝑁 WWalksN 𝐺 )