Theorem 0trl

Description: A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 7-Jan-2021) (Revised by AV, 30-Oct-2021)

	Ref	Expression
Hypothesis	0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	0trl	⊢ ( 𝐺 ∈ 𝑈 → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	0wlk	⊢ ( 𝐺 ∈ 𝑈 → ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
3	2	anbi1d	⊢ ( 𝐺 ∈ 𝑈 → ( ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ∅ ) ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ Fun ◡ ∅ ) ) )
4		istrl	⊢ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ∅ ) )
5		funcnv0	⊢ Fun ◡ ∅
6	5	biantru	⊢ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ↔ ( 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ∧ Fun ◡ ∅ ) )
7	3 4 6	3bitr4g	⊢ ( 𝐺 ∈ 𝑈 → ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ 𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )