So I want to prove that any one-to-one function of minimal sufficient statistic is also minimal sufficient. Here is my proof:

Let $T$ be a minimal sufficient statistic and $f$ is a one-to-one function such that $S=f(T)$. Since $T$ is minimal sufficient, then for any other sufficient statistic $T'$, $T=h(T')$ for some function $h$.

$\Rightarrow S=f(T)=f(h(T'))=g(T')$, where $g=f \circ h$. So $S$ is a function of $T'$, therefore $S$ is a minimal sufficient.

I'm not sure if this proof is correct because I'm not using the fact that $f$ is a one-to-one function. Can anyone give me some explanation or figure out which part is wrong?