I have a model with two endogenous variables X and Y, and a set of parameters that I denote by $\omega$. Equilibrium $X^{*}(\omega)$ and $Y^{*}(\omega)$ in this model is the solution to:
\begin{align*} F(X,Y;\omega)=0 \\ H(X,Y;\omega)=0 \end{align*}
I would like to know whether this equilibrium is unique (given parameters). I can prove that, if X is exogenously given, $Y^{*}(\omega, X)$ is uniquely determined. Similarly, I can prove that, if Y is exogenously given, $X^{*}(\omega, Y)$ is uniquely determined. Can I leverage these two results to show that the equilibrium with two endogenous variables is unique? I was thinking about proving it by contradiction, but not sure if it makes sense.
Thanks!