I have been trying to prove that implicit function theorem implies the inverse function theorem. Be $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\det[DF(x_0)]\neq 0$ for $x_0 \in \mathbb{R}^n$. Be $H: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined as follows:
$$H(x,y)= F(x)-y \quad \forall (x,y) \in \mathbb{R}^n \times \mathbb{R}^n . $$
We can use the implicit function theorem to see that exist $A \subset \mathbb{R}^n \times \mathbb{R}^n$ an open set that has $(x_0 , F(x_0))$ as an element, $B\subset \mathbb{R}^n$ an open set that has $F(x_0)$ as an element, and a function $f : B \rightarrow \mathbb{R}^n$ with: $$ A \cap \{ (x, y) : H(x,y)= 0 \}= \{ ( f(y), y) : y \in B \}. $$
It seems that $f$ is the inverse function of $F$, but I can't find a way to define the domain in which $F$ is bijective. Is there any tip?