Theorem 2trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
	2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
	2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
	2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
	2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
	2wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	2wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
Assertion	2trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2		2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3		2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4		2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5		2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6		2wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7		2wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8		2trld.n	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
9	1 2 3 4 5 6 7	2wlkd	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10	1 2 3 4 5	2wlkdlem7	⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) )
11		df-3an	⊢ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) ↔ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) ∧ 𝐽 ≠ 𝐾 ) )
12	10 8 11	sylanbrc	⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) )
13		funcnvs2	⊢ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐽 ≠ 𝐾 ) → Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
14	12 13	syl	⊢ ( 𝜑 → Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
15	2	cnveqi	⊢ ◡ 𝐹 = ◡ ⟨“ 𝐽 𝐾 ”⟩
16	15	funeqi	⊢ ( Fun ◡ 𝐹 ↔ Fun ◡ ⟨“ 𝐽 𝐾 ”⟩ )
17	14 16	sylibr	⊢ ( 𝜑 → Fun ◡ 𝐹 )
18		istrl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
19	9 17 18	sylanbrc	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )