Mellin Transform of Dirac Delta Function by Function

Theorem
Let $f: \R \to \R$ be a function.

Let $c \in \R_{>0}$ be a positive constant real number.

Let $\map {\delta_c} t$ be the Dirac delta function.

Let $\MM$ be the Mellin transform.

Then:
 * $\map {\MM \set {\map {\delta_c} t \map f t} } s = c^{s - 1} \map f c$