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 $\mathbb{Q}$-factorial variety with log canonical singularities, and $Z \subseteq X$ be a subvariety. Then the set $$\{E \mid E {\rm{~is~ an~ exceptional~ divisor~ of~ some~ resolution~ }}Y \to X, {\rm ~such ~ that~} f(E)=Z {\rm~and~} discrepancy(K_X, E) \leq 1 \}$$ is finite.

Any suggestion for the proof or references is welcome!!