Suppose the equation $f\left(x,y\right)=0$, with $x\in I_{1}$ and $y\in I_{2}$, $I_{1}$ and $I_{2}$ being open intervals. Additionally, consider that the conditions required to apply the Implicit Function Theorem (IFT) are verified for all $\left(x_{0},y_{0}\right)\in I_{1}\times I_{2}$. Hence, we can conclude that in a neighborhood containing the point $\left(x_{0},y_{0}\right)$, the equation $f\left(x,y\right)=0$ defines implicitly $y$ as a function of $x$.
And my question is: Since the conditions of IFT hold for all $\left(x_{0},y_{0}\right)\in I_{1}\times I_{2}$, is it true that the equation $f\left(x,y\right)=0$ defines implicitly $y$ as a function of $x$ with the domain of this implicit function being $I_{1}$?