Cowen-Engeler Lemma

Theorem
Let $X$ and $Y$ be sets.

Let $M$ be a set of mappings from subsets of $X$ to $Y$.

Thus each element of $M$ is a partial mapping from $X$ to $Y$.

Define a mapping $\Phi: X \to \mathcal P \left({Y}\right)$ thus:


 * $\Phi \left({x}\right) = \left\{{f \left({x}\right): f \in M \text{ and } x \in \operatorname{Dom} \left({f}\right) }\right\}$

That is:
 * $\Phi \left({x}\right) = \left\{{y \in Y: \exists f \in M: \left({x, y}\right) \in f}\right\}$

Suppose that the following hold:

Then there exists an element of $M$ with domain $X$.

Proof
Let $\operatorname{Fin} \left({X}\right)$ be the set of finite subsets of $X$.

For each $S \in \operatorname{Fin} \left({X}\right)$, let $\Gamma_S = \left\{{f \in M: S \subseteq \operatorname{Dom} \left({f}\right)}\right\}$.

By $(2)$, $\Gamma_S$ is non-empty for each $S \in \operatorname{Fin} \left({X}\right)$.

For each $S \in \operatorname{Fin} \left({X}\right)$, $\Gamma_S \cap \Gamma_T = \Gamma_{S \cup T}$:

If $f \in \Gamma_{S \cup T}$, then $S \cup T \subseteq \operatorname{Dom} \left({f}\right)$.

Then $S \subseteq \operatorname{Dom} \left({f}\right)$ and $T \subseteq \operatorname{Dom} \left({f}\right)$, so $f \in \Gamma_S$ and $f \in \Gamma_T$.

Thus $f \in \Gamma_S \cap \Gamma_T$.

If $f \in \Gamma_S \cap \Gamma_T$, then $f \in \Gamma_S$ and $f \in \Gamma_T$.

Thus $S \subseteq \operatorname{Dom} \left({f}\right)$ and $T \subseteq \operatorname{Dom} \left({f}\right)$.

Thus $S \cup T \subseteq \operatorname{Dom} \left({f}\right)$.

So $f \in \Gamma_{S \cup T}$.

Thus $\left\{ {\Gamma_S: S \in \operatorname{Fin} \left({X}\right)}\right\}$ has the finite intersection property.

Then by the corollary to the Ultrafilter Lemma, there is an ultrafilter $\mathcal U$ on on $M$ which includes $\left\{ {\Gamma_S: S \in \operatorname{Fin} \left({X}\right) }\right\}$ as a subset.

We will now construct a mapping $\phi: X \to Y$.

Fix $x \in X$.

Note that by the definitions of $\Phi$ and $\Gamma$:
 * $\Phi \left({x}\right) = \left\{ {f \left({x}\right): f \in \Gamma_{\left\{{x}\right\} } }\right\}$

For each $y \in \Phi \left({x}\right)$, let $K_y = \left\{ {f \in \Gamma_{\left\{{x}\right\} } : f \left({x}\right) = y }\right\}$.

Then $\left\{ {K_y : y \in \Phi \left({x}\right) }\right\}$ is a partition of $\Gamma_{\left\{{x}\right\} }$.

Since $\left\{{x}\right\} \in \operatorname{Fin} \left({X}\right)$:
 * $\Gamma_{\left\{{x}\right\} } \in \mathcal U$

By $(1)$, $\Phi \left({x}\right)$ is finite.

Thus by Component of Finite Union in Ultrafilter, there is exactly one $y \in \Phi \left({x}\right)$ such that $K_y \in \mathcal U$.

We define $\phi \left({x}\right)$ to be that $y$.

By the definition of $K_y$, for each $x \in X$ we have:


 * $\left\{ {f \in \Gamma_{\left\{{x}\right\} } : f \left({x}\right) = \phi \left({x}\right) }\right\} \in \mathcal U$

We must show that $\phi \in \mathcal M$.

Let $S$ be a finite subset of $X$.

Let $\displaystyle \Psi = \bigcap_{x \mathop \in S} \left\{ {f \in \Gamma_{\left\{{x}\right\} } : f \left({x}\right) = \phi \left({x}\right) }\right\}$.

Since:
 * $\left\{ {f \in \Gamma_{\left\{{x}\right\} } : f \left({x}\right) = \phi \left({x}\right) }\right\} \in \mathcal U$ for each $x \in X$
 * $S$ is finite
 * $\mathcal U$ is a filter

it follows that:
 * $\Psi \in \mathcal U$

Thus $\Psi \ne \varnothing$.

Choose any $f \in \Psi$.

Then $f \left({x}\right) \in \Gamma_{\left\{{x}\right\} }$ for each $x \in S$.

So $f \in M$ and $S$ is a finite subset of the domain of $f$.

Since $M$ has finite character, $f \restriction S \in M$.

Furthermore, by the definition of $\Psi$:
 * $f \left({x}\right) = \phi \left({x}\right)$

for each $x \in S$.

Thus:
 * $\phi \restriction S = f \restriction S \in \phi$

So we see that $\phi \restriction S \in M$ for each finite subset $S$ of $X$.

Since $M$ has finite character, $\phi \in M$.