Does the Implicit mapping theorem imply the inverse mapping theorem? Does the Implicit mapping theorem  imply the inverse mapping theorem?
 A: Yes. 
The implicit function theorem implies the inverse function theorem.
Consider $f : \mathbb{R}^3 \to \mathbb{R}^3$, as an example. Saying that $f$ has an inverse is precisely saying that the system
$$f1(x_1, x_2, x_3) = u_1$$
$$f2(x_1, x_2, x_3) = u_2$$
$$f3(x_1, x_2, x_3) = u_3$$
determines the values of the $x_i$’s if you’re given the $u_i$’s. 
But this is a system of three equations in six unknowns, and one can apply the
implicit function theorem to answer this question. Of course, the result
is precisely the inverse function theorem, that one can do this if $D$f is
invertible.
Can you generalize?
The converse is also true, that is: the Implicit and Inverse Function Theorems are equivalent; so each theorem implies the other.  

Note: The implicit and inverse function theorems are sometimes referred to as the implicit and inverse mapping theorems, respectively.
A: This is not an answer, it is the converse.
Suppose you want to 'solve' the implicit system $\phi(x,y) = 0$, ie, given $y$, find an $x$ satisfying the equation.
Create the extended system
$$f((x,y)) = \pmatrix{ \phi(x,y) \\ y }$$
Noting that this is a 'square' system, and apply the inverse function theorem to the equation 
$$f((x,y)) = \pmatrix{u_1 \\ u_2}$$
This gives functions $(u_1,u_2) \mapsto x((u_1,u_2))$, $(u_1,u_2) \mapsto y((u_1,u_2))$ satisfying the equation. From the form of the equation, we have $y((u_1,u_2)) = u_2$, hence by setting $u_1 = 0$ we have 'solved' the equation $\phi(x((0,u_2)),u_2) = 0$.
In many situations, I prefer to apply Newton's method rather than use the inverse function theorem directly. They are equivalent, of course, but Newton's method makes some details explicit that only appear in the proof of the inverse function theorem. In particular, Newton's method shows that the solution can be computed using the Newton iteration $x_{n+1} = x_n -Df(x_n)^{-1} (f(x_n))$. This equation can often be used to make properties of the solution 'obvious' (eg, symmetry in the case of matrices, belonging to a closed space, etc.).
