Fn forcing ($\mathrm{Fn}(X,Y,\kappa )$)

The order $\mathrm{Fn}(X,Y,\kappa )$ consists of partial functions $p:X\rightharpoonup Y$ with $|\mathrm{dom}p|<\kappa$, ordered by reverse inclusion so that larger functions are stronger conditions.

The order $\mathrm{Fn}(X,Y,\omega )$ has the countable chain condition for forcing whenever $Y$ is countable. For an uncountable family of conditions, apply the Delta-system lemma to the finite domains. On their common finite root there are only countably many assignments, so thin to an uncountable family agreeing on that root. Any two conditions now have a compatible union. For binary Cohen forcing, there are only finitely many root assignments.

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.