I'm trying to prove the following result:
Let $r:X\longrightarrow Y$ be a relation and $\mathcal{U}$ and $\mathcal{V}$ be ultrafilters such that $r^{-1}V \in\mathcal{U}$ for all $V\in\mathcal{V}$. There exists an ultrafilter $\mathcal{X}$ on $\Gamma_r = \{(x,y)\in X\times Y | xry\}$ for which one has $\pi_X(\mathcal{X}) = \mathcal{U}$ and $\pi_Y(\mathcal{X}) = \mathcal{V}$.
My guess was to put $C_{U,V} = \pi_X^{-1}(U)\cap\pi_Y^{-1}(V) = U\times V \cap\Gamma_r$ for all $U\in\mathcal{U}$ and $V\in\mathcal{V}$. Because of the assumption on $r$, these sets are never empty and satisfy the finite intersection property. Hence, they form a basis for a filter $\mathcal{X}$ on $\Gamma_r$. Obviously one has $\pi_X(\mathcal{X})=\mathcal{U}$ and $\pi_Y(\mathcal{X})=\mathcal{V}$.
Yet, I'm stuck at proving that this filter $\mathcal{X}$ is an ultrafilter. First of all, is this indeed an ultrafilter, and if not, is there one that satisfies the conditions? Any help would be appreciated!
Hint: A (proper) filter can be extended to an ultrafilter.