Proof of the implicit and inverse function theorems

[r4nd0m]

Today I revised my knowledge from multivariable calculus and I found that I couldn't remember the proofs of these two theorems. Then I looked in Rudin, and everything was clear.
Except one thing, which probably made me forgot the proofs. There are two weird functions in these two proofs: $$\phi(x) = x + A^{-1} (y-f(x))$$ and $$F(x,y) = (f(x,y),y)$$.
I can see how they are used and that it really works, but I don't really understand what do the functions really say, if you know what I mean. How did someone figure out that these are the right functions we should use in the proof?

 

[fresh_42]

Hard to tell what you exactly mean without seeing the proofs. However, $F(x,y)=(f(x,y),y)$ is a standard construction when dealing with the implicit function theorem, i.e. it is the implicit function theorem!

The first function is something like a derivative: $A= \dfrac{y-f(x)}{\phi(x)-x}$, so again a standard quotient to consider.