Theorem 1trld

Description: In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017) (Revised by AV, 22-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
	1wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 ”⟩
	1wlkd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	1wlkd.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	1wlkd.l	⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝐼 ‘ 𝐽 ) = { 𝑋 } )
	1wlkd.j	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ⊆ ( 𝐼 ‘ 𝐽 ) )
	1wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	1wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	1trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		1wlkd.p	⊢ 𝑃 = ⟨“ 𝑋 𝑌 ”⟩
2		1wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 ”⟩
3		1wlkd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
4		1wlkd.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
5		1wlkd.l	⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝐼 ‘ 𝐽 ) = { 𝑋 } )
6		1wlkd.j	⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → { 𝑋 , 𝑌 } ⊆ ( 𝐼 ‘ 𝐽 ) )
7		1wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
8		1wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
9	1 2 3 4 5 6 7 8	1wlkd	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10		funcnvs1	⊢ Fun ◡ ⟨“ 𝐽 ”⟩
11	2	cnveqi	⊢ ◡ 𝐹 = ◡ ⟨“ 𝐽 ”⟩
12	11	funeqi	⊢ ( Fun ◡ 𝐹 ↔ Fun ◡ ⟨“ 𝐽 ”⟩ )
13	10 12	mpbir	⊢ Fun ◡ 𝐹
14		istrl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
15	9 13 14	sylanblrc	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )