Local implicit function parameterization

Let $F:U\subseteq\mathbb{R}^{n+m}\to\mathbb{R}^m$ be $C^1$ , and suppose $(a,b)\in U$ satisfies $F(a,b)=0$ and $\det\left(\frac{\partial F}{\partial y}(a,b)\right)\neq 0.$

Corollary: There exist neighborhoods $A$ of $a$ and $B$ of $b$ and a $C^1$ function $g:A\to B$ such that the solution set of $F(x,y)=0$ in $A\times B$ is exactly the graph of $g$: $\{(x,y)\in A\times B:\ F(x,y)=0\}=\{(x,g(x)):\ x\in A\}.$

Connection to parent theorem: This is precisely the conclusion of the implicit function theorem , viewed as a geometric parameterization statement.