Properties of image of sub-vector bundle

Question

I have the following problem:

Let $E,F \rightarrow C$ vector bundles over $C$, with $C$ an algebraic curve of genus $g>1$ and $rank(E)=rank(F)+1$, let $ p: E \rightarrow F$ a surjective vector bundle morphism and let $D$ a proper sub-bundle of $E$.

Is $p(D)$ a vector bundle? if it is, how is the degree of $p(D)$ respect to the degree of $D$?

My first attempt was think that $deg( p(D)) = deg(p)deg(D)$ with $deg(p)$ the degree as a function, but i don't know if this makes sense.

Answer 1

$p(D)$ need not be a subbundle. As an example, take $E=O\oplus O, F=O$ and $p$ a projection to one of the factors. Let $D$ be a very negative line bundle. Then, you can find an inclusion of $D$ into $E$ so that the image is a subbundle. Clearly $p(D)$ is not a subbundle since the only subbundles of $O$ are $0$ or $O$.