Bases of Vector Space have Equal Cardinality

Theorem
Let $V$ be a vector space.

Let $X$ and $Y$ are bases of $V$.

Then $X$ and $Y$ are equinumerous.

Proof
We will first prove that there is an injection from $X$ to $Y$.

Let $\mathcal F$ be the set of all partial functions $f$ from $X$ to $Y$ such that:


 * $f$ is one-to-one.
 * If $(x,y) \in f$, then $x$ can be expressed as a finite sum of non-zero multiples of elements of $Y$, one of which is $y$.