Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties:

1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$.

2) $f_i(x_1, x_2, ..., x_n)=1$ if $x_i=1$.

3) $\frac{\partial f_i(x_1, x_2, ..., x_n)}{\partial x_i}>0$, for $0<x_i<1$

4) $\frac{\partial f_j(x_1, x_2, ..., x_n)}{\partial x_i}<0$, for $j\neq i$ and $0<x_i<1$

Fix $p_1, ..., p_n$, with $0<p_i<1$ for $i=1,...n$. I want to prove that there is a unique solution $x_1, ..., x_n$, with $0<x_i<1$ for $i=1,...n$, such that $f_i(x_1, x_2, ..., x_n)=p_i$ for all $i=1,...n$.

Are the listed properties enough? If not, what extra properties do I require of the functions $f_i$?