By Theorem DCM we know that $\complex{n}$ has dimension $n$. So by Theorem G we need only establish that the set $C$ is linearly independent or a spanning set. However, the hypotheses also require that $C$ be of size $n$. We assumed that $B=\set{\vectorlist{x}{n}}$ had size $n$, but there is no guarantee that $C=\set{A\vect{x}_1,\,A\vect{x}_2,\,A\vect{x}_3,\,\dots,\,A\vect{x}_n}$ will have size $n$. There could be some “collapsing” or “collisions.”
Suppose we establish that $C$ is linearly independent. Then $C$ must have $n$ distinct elements or else we could fashion a nontrivial relation of linear dependence involving duplicate elements.
If we instead choose to prove that $C$ is a spanning set, then we could establish the uniqueness of the elements of $C$ quite easily. Suppose that $A\vect{x}_i=A\vect{x}_j$. Then
\begin{align*}
A(\vect{x}_i-\vect{x}_j)&=A\vect{x}_i - A\vect{x}_j=\zerovector
\end{align*}
Since $A$ is nonsingular, we conclude that $\vect{x}_i-\vect{x}_j=\zerovector$, or $\vect{x}_i=\vect{x}_j$, contrary to our description of $B$.