Theorem 3trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
	3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
	3wlkd.s	⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
	3wlkd.n	⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
	3wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
	3wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	3wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	3trld.n	⊢ ( 𝜑 → ( 𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿 ) )
Assertion	3trld	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		3wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2		3wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3		3wlkd.s	⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
4		3wlkd.n	⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
5		3wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
6		3wlkd.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7		3wlkd.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8		3trld.n	⊢ ( 𝜑 → ( 𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿 ) )
9	1 2 3 4 5 6 7	3wlkd	⊢ ( 𝜑 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10	1 2 3 4 5	3wlkdlem7	⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V ) )
11		funcnvs3	⊢ ( ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V ) ∧ ( 𝐽 ≠ 𝐾 ∧ 𝐽 ≠ 𝐿 ∧ 𝐾 ≠ 𝐿 ) ) → Fun ◡ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
12	10 8 11	syl2anc	⊢ ( 𝜑 → Fun ◡ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
13	2	cnveqi	⊢ ◡ 𝐹 = ◡ ⟨“ 𝐽 𝐾 𝐿 ”⟩
14	13	funeqi	⊢ ( Fun ◡ 𝐹 ↔ Fun ◡ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
15	12 14	sylibr	⊢ ( 𝜑 → Fun ◡ 𝐹 )
16		istrl	⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
17	9 15 16	sylanbrc	⊢ ( 𝜑 → 𝐹 ( Trails ‘ 𝐺 ) 𝑃 )