Anomalous magnetic moment of the electron - integration problem

In Schwartz's QFT book (eqn 17.31), to find the anomalous magnetic moment of the electron from the form factors, near the end of the calculation the following integral needs to be evaluated:

$$F_{2}(0) = \frac{\alpha}{\pi} \int_{0}^{1} dz \int_{0}^{1} dy \int_{0}^{1} dx \frac{\delta(x+y+z-1) z}{1-z}$$

which gives the result

$$F_{2}(0) = \frac{\alpha}{\pi} \int_{0}^{1} dz \int_{0}^{1-z} dy \frac{z}{1-z}$$

My question is, how do you fill in the gap of this calculation? I am not sure how to use the delta function on a definite integral, especially when there is more than 1 variable.

My attempt has been to integrate over $$x$$ using the delta function to set $$x = 1-y-z$$, and then evaluate this at the limits $$0 \: \& \: 1$$, which gives $$y = 1-z$$ for the upper limit and $$y=z=0$$ for the lower one. But I'm not sure why this factor appears in the limit of the integral over $$y$$ and not elsewhere. Any tips on how to approach these kinds of integrals would be appreciated, as they seem to be important when using Feynman parameter.

Hint: Well, the condition on $$x$$ is $$x= 1\!-\!y\!-\!z \in[0,1]$$, so the condition on $$y$$ is $$y\!+\!z \in[0,1]\Leftrightarrow y\in [-z,1\!-\!z]$$. Together with the condition $$y\in[0,1]$$, we get $$y\in [-z,1\!-\!z]\cap [0,1]=[0,1\!-\!z]$$, which is the displayed integration limits.