Charts on a Manifold

Question

Let $f^{1},\cdots ,f^r,\Phi ^1,\cdots , \Phi ^{n-r}$ be functions of class $C^{(1)}$ on an open set $D$. suppose that $F=(f^1\mid S,\cdots ,f^r\mid S)$ is a coordinate system for $S$, that $S=\lbrace x\in D:\Phi (x)=0\rbrace $, and that $D\Phi (x)$ has rank $n-r $ for every $x\in D$. Show that $x_{0}\in S$ has a neighborhood $U$ such that $(f^{1},\cdots ,f^r,\Phi ^1,\cdots , \Phi ^{n-r})$ restricted to $U$ is a coordinate system for $U$. So far I have tried to show S is an r manifold and being open subset of S then U is also an r manifold. but I do not succeeded to show that the given system is a coordinate system for U.