If $Y \sim \operatorname{Gamma}(a+1, 1), U_0 \sim \operatorname{Unif}(0, 1), U = U_0^\frac1a$, why $YU \sim \operatorname{Gamma} (a, 1)$? Given random variables like below:

$Y \sim \operatorname{Gamma}(a + 1, 1)$
$U_0 \sim \operatorname{Unif}(0,1)$
$U = U_0^\frac1a$

If $Y$  and $U_0$ is independent,
How can I proof $X=YU \sim \operatorname{Gamma}(a, 1)$ ?
I tried to solve this problem with theorem $f_{U,V}(u,v) = f_{X,Y}(h_1(u,v), h_2(u,v))|J|$
But I'm confusing what should be preimage.
 A: Find the moment generating function of $YU$. Use independence to make it equal to MGF of $Y$ times MGF of $U$. Look up the gamma MGF and uniform MGF and then times them together, and see that it is also gamma.
A: Again, like in If $Y\sim\operatorname{Beta}(a,1-a)$ and $Z\sim\operatorname{Exp}(1)$, then $YZ\sim\operatorname{Gamma}(0,1)$? we will be using Mellin transform to get image of the product, using the fact, that image of the product is the product of Mellin images. For $U(0,1)^{\frac{1}{a}}$ we could easily reconstruct PDF
$$
PDF_X(x) = a x^{a-1} 1_{0<x<1}
$$
Its Mellin image is
$$
M(X) = \frac{a}{s+a-1}
$$
For Gamma-distribution
$$
M(Y) = \frac{\Gamma(s+a-1+1)}{\Gamma(a+1)}
$$
Therefore
$$
M(XY) = \frac{\Gamma(s+a-1+1)}{s+a-1} \frac{a}{\Gamma(a+1)} = \frac{\Gamma(s+a-1)}{\Gamma(a)}  
$$
using well-known property of Gamma-function. Which is obviously could be transformed back to
$$
PDF(x|XY) = M^{-1}\{\frac{ \Gamma(s+a-1) }{\Gamma(a)} \} = \frac{ \exp(-x) x^{a-1} }{\Gamma(a)} 1_{x>0}
$$
