I am formulating the elementary simplex in $\mathbb R^n$ as $\{(x_1,..,x_n):0\leq x_i\leq 1,\,\sum_{i=1}^n x_i<1\}$. Correspondingly, its volume $V$ should be equal to ($\mathbb 1_A$ is the indicator function over $A$): $$ V=\int_0^1\text d x_1..\int_0^1\text dx_n \mathbb 1_{\sum_{i=1}^n x_i<1}. $$
Normally, this integral is written under the form: $$ V=\int_0^1\text d y_1..\int_0^{y_{n-1}}\text dy_n=\frac 1{n!}.$$
I am looking for the change of variable leading from the first to the second form. I attempted $y_i:=1-\sum_{j=i}^n x_j$, but indeed did not work.
How would you do this?