Let $X,Y$ be two affine varieties (irreducible zero loci of polynomials over algebraically closed field). Let $\phi : X\rightarrow Y$ be a finite morphism (i.e. K[X] is finitely generated $\phi ^*(K[Y])-module$ ). I want to show that there exists a natural number $n$ such that $|\phi^{-1}(p)|<n$ for every $p\in Y$.
Note that $K[X]$ denotes the affine coordinate ring of $X$.
I expect that this $n$ is the same as the number of finitely many generators but I am not able to deduce that. I know that fibers are finite (since finite maps preserve strict inclusions). Would anyone kindly provide a hint for this?