 Solution

Let $A\subseteq \mathrm{Fn}(I,J)$ be uncountable. Apply the Delta-system lemma to the finite sets $\mathrm{dom}p$ for $p\in A$. After passing to an uncountable subset $A^{\prime }$, there is a fixed finite root $R$ such thatfor distinct $p,q\in A^{\prime }$. Because $J$ is countable and $R$ is finite, there are only countably many functions $R\to J$. A further uncountable subset $A^{\prime \prime }\subseteq A^{\prime }$ therefore has the same restriction to $R$.

Any two conditions in $A^{\prime \prime }$ agree on the intersection of their domains, so their union is a common stronger condition. Thus every uncountable family contains two compatible conditions, and no uncountable antichain exists. Therefore