Changing order of integration in triple integral

Change the order of integration in the triple integral $$\int_0^1 \int_0^x \int_0^y f(x,y,z) \,dz ~dy~dx$$ to $dz~dx~dy$

Attempt: Here, since $z$ remains in the same place I think it is enough to consider only what happens in the $xy$ plane. So the bounds will become $0 \leq y \leq 1$, $y \leq x \leq 1$ and $0 \leq z \leq y$. However, my intuition tells me that this is not correct since if we change the order of $x$ and $y$ then this will affect $z$ as well. So is my intuition correct? If yes then how can I approach this problem?

Just focus on $x$ and $y$, as $z$ is not changing its position. Then,

Original: $$\color{red}{0\le x\le1,\space 0\le y\le x},\space 0\le z\le y$$ Alternative: $$\color{red}{y\le x\le1,\space 0\le y\le 1},\space 0\le z\le y$$

So you can change the integral to $$\int_0^1 \int_y^1 \int_0^y f(x,y,z) \,dz~dx~dy$$