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 e. U -> ( (/) ( Trails ` G ) P <-> P : ( 0 ... 0 ) --> V ) )

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	|- V = ( Vtx ` G )
2	1	0wlk	|- ( G e. U -> ( (/) ( Walks ` G ) P <-> P : ( 0 ... 0 ) --> V ) )
3	2	anbi1d	|- ( G e. 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 e. U -> ( (/) ( Trails ` G ) P <-> P : ( 0 ... 0 ) --> V ) )