Equivalence of Definitions of Arborescence

Theorem
The following definitions of arborescence are equivalent:

Proof
Let $G = (V, A)$ be a directed graph.

Let $r \in V$.

Definition 1 implies Definition 3
Suppose that $G$ is an $r$-arborescence by Definition 1.

Let $v \in V$ with $v \ne r$.

Then there is exactly one directed walk $w$ from $r$ to $v$.

Since $v \ne r$, either $w = (r, v)$ or there exists some vertex $m \in V$ such that $w = (r, \dots, m, v)$.

Thus $v$ is the final vertex of the arc $rv$ or the arc $mv$.

Suppose for the sake of contradiction that $v$ is the final vertex of distinct arcs $xv, yv$.

Then there are directed walks $w_1$ and $w_2$ from $r$ to $x$ and $r$ to $y$, respectively.

But appending $v$ to $w_1$ and to $w_2$ yields distinct directed walks from $r$ to $v$, contradicting the fact that there is exactly one such directed walk.

Thus $v$ is the final vertex of exactly one arc.

Suppose for the sake of contradiction that $r$ is the final vertex of an arc $xr$.

By Definition 1, there is a directed walk $w$ from $r$ to $x$.

But then $w$ appended to $w$ is a directed walk from $r$ to $x$ which is not equal to $w$, contradicting Definition 1.

Thus we conclude that $r$ is not the final vertex of any arc.

It follows immediately from Definition 1 that there is a directed walk from $r$ to each vertex $v \ne r$.

Thus $G$ is an arborescence by Definition 3.