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Let $F$ be family of function for which $f,g \in F \rightarrow f \subseteq g$ or $g \subseteq f$. I need to prove that $\cup F$ is a function.

The result is very obvious, but how the proof should go is not. For example, is it allowed to pick the function with the greatest cardinality in the family and then show that $\cup F$ is that function? What kind of proof technique should be used?

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First of all, note that your family $F$ does not necessarily have a maximal element: it could be an ever-increasing chain of functions.

The proof, however, is rather straightforward using what it means for $\cup F$ to be a function.

Take $(x,y), (x',y') \in \cup F$ with $x = x'$. We must show that $y = y'$. Because $(x,y) \in \cup F$, there is an $f \in F$ with $(x,y) \in f$ and similarly there is a $g \in F$ with $(x,y') \in g$. By the assumption on $F$, $f \subseteq g$ or $g \subseteq f$; without loss of generality we'll assume $f \subseteq g$. Then $(x, y) \in g$ as well and therefore, because $g$ is a function, $y = y'$.

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