Rank of the Pushforward under a Finite Surjective Morphism from an Integral Curve to a Smooth Curve

Let $X$ be an integral curve, $Y$ a smooth curve, and $f : X \to Y$ a finite surjective morphism. Given a vector bundle $E$ on $X$, what is $\operatorname{rk}(f_{*}E)$? Is it true that

$$ \operatorname{rk}(f_{*}E)=\operatorname{rk}(E)\deg(f)? $$

If so, could you give me a proof or indicate a reference for this result?

I encountered this situation in the Beauville--Narasimhan--Ramanan correspondence for Higgs bundles.

Here is my attempt. Since $f$ is finite and surjective, it is dominant. Since $Y$ is a smooth curve, for every $y \in Y$ the local ring $\mathcal{O}_{Y,y}$ is a discrete valuation ring. Since $X$ is integral and $f$ is dominant, for $x \in X$ with $f(x)=y$, the map

$$ \mathcal{O}_{Y,y} \longrightarrow \mathcal{O}_{X,x} $$

is injective, and $\mathcal{O}_{X,x}$ is torsion-free as an $\mathcal{O}_{Y,y}$-module. Since a torsion-free module over a discrete valuation ring is flat, this should imply that $f$ is flat.

Thus $f$ is finite and flat, hence finite locally free. In particular,

$$\deg(f)$$.

What I am not completely sure about is the final step for an arbitrary vector bundle $E$ on $X$. How does one rigorously deduce that $f_{*}E$ is locally free on $Y$ and that

$$\deg(f)\operatorname{rk}(E)?$$

Is there a standard reference for this result?