Theorem spthonisspth

Description: A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 18-Jan-2021)

	Ref	Expression
Assertion	spthonisspth	⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	spthonprop	⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ∧ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) )
3		simp3r	⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ∧ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )
4	2 3	syl	⊢ ( 𝐹 ( 𝐴 ( SPathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 )