Let $\pi: X \to B$ be an elliptic surface with a section $\sigma$. Assume that $\pi, X, B$ are all smooth and that $X,B$ are projective.
Let $\mathcal M(r,d)_{X/B}$ be the relative moduli space of stable vector bundles of rank r on fibers with degree d.
I have seen a few times using ZMT (Zariski main theorem) used the following way:
Claim: Assume there is an injective map $\phi:X \to \mathcal M(r,d)_{X/B}$. Then by ZMT, $\phi$ is an isomorphism.
I am looking for references for the above usage of ZMT. I looked into here and here but could not a version or a ZMT corollary suitable for the claim.
Thank you for any hint or reference.