Show that the equation $x^4 + y^4 -2xy=0$ defines a function $\phi(x)=y$ in a neighborhood of $(1,1)$.
Proof. Let $f(x,y)=x^4+y^4-2xy$. Clearly $f\in C^{\infty}$, and $f$ is continuously differentiable in $A\times B$, which contains the point $(1,1)$. Therefore there exists open sets such that $1_x \in A$, $1_y \in B$ where $f(1,1)=1+1-2=0$. Also the Jacobian of $f$ with respect to $y$ ( In this case it is $D_2 f = 4y^3 - 2x \neq 0$ ) has non-zero determinant when $y\in B$. Therefore the implicit mapping theorem states that for each $x\in A$ there is a unique $\phi(x) \in B$, that is we can solve $y$ in terms of $x$. There is a function $\phi(x)=y$ defined near $(1,1)$.
Is this an accurate application of the implicit mapping theorem?