Theorem wlkoniswlk

Description: A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017) (Revised by AV, 2-Jan-2021)

	Ref	Expression
Assertion	wlkoniswlk	⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	wlkonprop	⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) )
3		simp31	⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = 𝐵 ) ) → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
4	2 3	syl	⊢ ( 𝐹 ( 𝐴 ( WalksOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )