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	$⊢ G \in U \to (\emptyset Trails (G) P \leftrightarrow P : (0 \dots 0) ⟶ V)$

Proof

Step	Hyp	Ref	Expression
1		0wlk.v	$⊢ V = Vtx (G)$
2	1	0wlk	$⊢ G \in U \to (\emptyset Walks (G) P \leftrightarrow P : (0 \dots 0) ⟶ V)$
3	2	anbi1d	$⊢ G \in U \to ((\emptyset Walks (G) P \land Fun \emptyset^{-1}) \leftrightarrow (P : (0 \dots 0) ⟶ V \land Fun \emptyset^{-1}))$
4		istrl	$⊢ \emptyset Trails (G) P \leftrightarrow (\emptyset Walks (G) P \land Fun \emptyset^{-1})$
5		funcnv0	$⊢ Fun \emptyset^{-1}$
6	5	biantru	$⊢ P : (0 \dots 0) ⟶ V \leftrightarrow (P : (0 \dots 0) ⟶ V \land Fun \emptyset^{-1})$
7	3 4 6	3bitr4g	$⊢ G \in U \to (\emptyset Trails (G) P \leftrightarrow P : (0 \dots 0) ⟶ V)$

Metamath Proof Explorer

Theorem 0trl

Proof