This essentially follows from Claim 2.3.1, p. 39 of

J. Kollár, S. Kovács: *[Singularities of the Minimal Model Program][1]*, Cambridge Tracts in Mathematics **200** (2013).

The Authors prove the result for $m=1$, but the same proof works without modifications for every $m \geq 1$. Compare also with Claim 2.3.2 (same page), Proposition 2.17 (p. 48) and Proposition 2.18 (p. 49). 

The last result, applied with $S=X, \, p_X=\text{id}_X$, $\Delta=\Delta_Y=\emptyset$, gives precisely the statement you need.
 

 


  [1]: https://www.cambridge.org/core/books/singularities-of-the-minimal-model-program/BFD87F262E5174A425F12AD85270E0AA