# Understanding a lemma for Implicit Function Theorem

The following lemma is a key for Implicit Function Theorem, given in Topology and Geometry by Bredon; but I am unable to see what the theorem is stating, what its conditions imply geometrically. I tried in small dimension case ($$n=m=1$$), but could not proceed.

Let $$\xi\in\mathbb{R}^n$$ and $$\eta\in\mathbb{R}^m$$ be given. Let $$f:\mathbb{R}^n\times\mathbb{R}^m\rightarrow\mathbb{R}^m$$ be $$C^1$$ and put $$f=(f_1,\ldots, f_m)$$. Assume that $$f(\xi,\eta)=\eta$$ and that all the following partial derivatives vanish at $$(\xi,\eta)$$: $$\frac{\partial f_i}{\partial y_j}(\xi,\eta)=0$$ where $$x_1,\ldots, x_n$$ are coordinates in $$\mathbb{R}^n$$ and $$y_1,\ldots,y_n$$ are in $$\mathbb{R}^m$$. Then there exists real numbers $$a>0$$ and $$b>0$$ such that there exists a unique function $$\phi:A\rightarrow B$$ where $$A=\{x\in\mathbb{R}^n \,:\, |x-\xi|\le a\}$$ and $$B=\{y\in\mathbb{R}^m \,:\, |y-\eta|\le b\}$$, such that $$\phi(\xi)=\eta \hskip5mm \mbox{ and } \hskip5mm \phi(x)=f(x,\phi(x)) \hskip5mm \forall x\in A.$$ Moreover $$\phi$$ is continuous.

Can one explain what the lemma is saying?

The function $$f$$ is taken as $$f(\xi,\eta)=\eta$$; this looks like a projection map, but it is not overall projection map (since, only at point $$(\xi,\eta)$$, value is second component $$\eta$$). So, I did not get the importance of this assumption.

Further, the $$m$$ partial derivatives of $$m$$ components of $$f$$ are taken $$0$$, so in the $$Df(\xi,\eta)$$, the last $$m\times m$$ part is fully $$0$$; When one tries to put condition on an $$m\times m$$ matrix, the basic conditions come are whether it is singular or not; but he is putting more than singular condition. I did not get reason behind this.

What the lemma is saying is that if $$F(\xi, \eta) = \eta$$ and $$D_yF(\xi, \eta) = 0$$, then the equation $$F(x, y) = y$$ defines $$y$$ implicitly as a continuous function $$y = \phi(x)$$ of $$x$$ for $$x$$ near $$\xi$$.
Really the lemma is just the implicit function theorem applied to $$G(x, y) = F(x, y) - y$$.