In Mathematical Analysis by Andrew Browder, he states the implicit function theorem as follows:
Let $\Omega$ be an open set in $\mathbb{R}^{n+m}$, and let $\textbf{f}: \Omega \rightarrow \mathbb{R}^m$ be a mapping of class $C^r, r \geq 1$. Let $\mathbf{p}=(\mathbf{a}, \mathbf{b})$, and let $\Sigma$ be the "level surface" of $\mathbf{f}$ through $\mathbf{p}$:
$$\Sigma=\{\mathbf{q}\in \Omega: \mathbf{f}(\mathbf{q})=\mathbf{f}(\mathbf{p})\}.$$ Define $S \in\mathscr{L}(\mathbb{R}^n, \mathbb{R}^m)$ by $S(\mathbf{h})=\mathrm{d}\mathbf{f}_\mathbf{p}(\mathbf{h},\mathbf{0})$ and $T \in \mathscr{L}(\mathbb{R}^m)$ by $T(\mathbf{k})=\mathrm{d}\mathbf{f}_\mathbf{p}(\mathbf{0},\mathbf{k}).$ Suppose that $T$ iis invertible. Then there exists a neighborhood $U$ of $\mathbf{p} \in \mathbb{R}^{n+m}$ a neighborhood $W$ of $\mathbf{a}$ in $\mathbb{R}^n$, and a mapping $\mathbf{g}: W \rightarrow \mathbb{R}^m$ which is of class $C^r$, such that
$$\Sigma \cap U=\{(\mathbf{s},\mathbf{g}(\mathbf{s}): \mathbf{s} \in W \}.$$ Furthermore $\mathrm{d}\mathbf{g}_{\mathbf{a}} = -T^{-1}S$.
This I thought was a rather convoluted statement, so I looked the statement up on Wikipedia, which states:
Let $f: \mathbb{R}^{n+m} \to \mathbb{R}^m$ be a continuously differentiable function, and let $\mathbb{R}^{n+m}$ have coordinates $(\textbf{x}, \textbf{y})$. Fix a point $(\textbf{a}, \textbf{b}) = (a_1,\dots,a_n, b_1,\dots, b_m)$ with $f(\textbf{a}, \textbf{b}) = > \mathbf{0}$, where $\mathbf{0} \in \mathbb{R}^m$ is the zero vector. If the Jacobian matrix (this is the right-hand panel of the Jacobian matrix shown in the previous section): $$J_{f, \mathbf{y}} (\mathbf{a}, \mathbf{b}) = \left [ \frac{\partial f_i}{\partial y_j} (\mathbf{a}, \mathbf{b}) \right ]$$ is invertible, then there exists an open set $U \subset \mathbb{R}^n$ containing $\textbf{a}$ such that there exists a unique continuously differentiable function $g: U \to \mathbb{R}^m$ such that $ g(\mathbf{a}) = \mathbf{b}$, and $ f(\mathbf{x}, g(\mathbf{x})) = \mathbf{0} ~ \text{for all} ~ \mathbf{x}\in U$. Moreover, denoting the left-hand panel of the Jacobian matrix shown in the previous section as: $$ J_{f, \mathbf{x}} (\mathbf{a}, \mathbf{b}) = \left [ \frac{\partial f_i}{\partial x_j} (\mathbf{a}, \mathbf{b}) \right ], $$ the Jacobian matrix of partial derivatives of $g$ in $U$ is given by the matrix product: $$ \left[\frac{\partial g_i}{\partial x_j} (\mathbf{x})\right]_{m\times n} =- \left [ J_{f, \mathbf{y}}(\mathbf{x}, g(\mathbf{x})) \right ]_{m \times m} ^{-1} \, \left [ J_{f, \mathbf{x}}(\mathbf{x}, g(\mathbf{x})) \right ]_{m \times n} $$
My question is: how are they the same statement? I'm mostly confused how to prove that the $\Sigma \cap U$ condition in the Browder definition is equivalent to the "$ g(\mathbf{a}) = \mathbf{b}$, and $ f(\mathbf{x}, g(\mathbf{x})) = \mathbf{0} ~ \text{for all} ~ \mathbf{x}\in U$" (or really $W$) condition on Wikipedia.