I feel the following fact has been used in many argument in algebraic geometry, but I was not be able to prove it or find the precise reference:

Let $X$ be a normal variety, and $Z \subseteq X$ be a subvariety. Then the set $$\{E \mid E {\rm{~is~ an~ exceptional~ divisor~ of~ some~ resolution~ }}Y \to X, {\rm ~such ~ that~} f(E)=Z \}$$ is finite (I guess $E$ is called geometric valuation).

Any suggestion for the proof or references is welcome!!