Prove that an isomorphism produces a basis of W

I think I may have the answer for this one, but I could use a second opinion.

Suppose $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

Now suppose $\displaystyle \psi:V\rightarrow W$ is an isomorphism and $\displaystyle \{ v_1,...,v_n\}$ is a basis of $\displaystyle V$. Prove that $\displaystyle \{\psi(v_1),...,\psi(v_n)\}$ is a basis of $\displaystyle W$.

Here is what I have so far:

Let $\displaystyle \{w_1,...,w_n\}$ be a basis for $\displaystyle W$ where

$\displaystyle w_1=\psi(v_1),...,w_n=\psi(v_n), \forall n>0$.

Then we consider the map

$\displaystyle \psi(x)=x_1w_1+x_2w_2+...+x_nw_n$ where $\displaystyle x=x_1v_1+x_2v_2+...+x_nv_n\in V$.

Since the bases are inverse to each other, we can conclude that $\displaystyle \{\psi(v_1),...,\psi(v_n)\}$ is a basis of $\displaystyle W$.

I don't know if I have any major holes in my proof, so if you see any, please let me know.