Theorem pthontrlon

Description: A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021)

	Ref	Expression
Assertion	pthontrlon	⊢ ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2	1	pthsonprop	⊢ ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) )
3		simp3l	⊢ ( ( ( 𝐺 ∈ V ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝐹 ∈ V ∧ 𝑃 ∈ V ) ∧ ( 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ) → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )
4	2 3	syl	⊢ ( 𝐹 ( 𝐴 ( PathsOn ‘ 𝐺 ) 𝐵 ) 𝑃 → 𝐹 ( 𝐴 ( TrailsOn ‘ 𝐺 ) 𝐵 ) 𝑃 )