I am trying to prove that the random variable:
$Z = 1-X-Y$
where $X\sim U[0,1]$ and $Y\sim U[0, 1-X]$
is $Z \sim U[0,1]$. I'm not sure this is even true, but it feels correct. I'm having trouble with the formalities of how to go about attempting to prove this, especially since $Y$ depends on $X$. Any guidance would be much appreciated.