Can someone please verify my proof and offer suggestions for improvement? I feel that my proof might have been a little hand-waving in showing that $\varphi$ is a bijection, and I feel that it is not too readable.
Some preliminaries: If $Q$ and $R$ are posets, $Q+R$ is the poset on $Q \cup R$. Similarly, if $A$ and $B$ are posets, $(x, y) \leq_{A \times B} (x', y') \iff x \leq_A x'$ and $y \leq_B y'$. If $P$ and $Q$ are posets, $P^Q$ is the poset consisting of all order preserving functions from $Q$ to $P$, where $f \leq g \iff f(x) \leq g(x)$ for all $x \in P$.
Let $P, Q, R$ be finite posets. Prove that $P^{Q+R} \cong P^Q \times P^R$.
Let $\varphi:P^{Q+R} \longrightarrow P^Q \times P^R$ be defined by $\varphi(f) = (f|_Q, f|_R)$. $\varphi$ is clearly a bijection. It remains to show that $\varphi$ is order preserving. Clearly, if $f, g \in P^{Q+R}$, then
\begin{eqnarray} f \leq g &\iff& f(x) \leq g(x) \ \ (\forall x \in Q+R) \\ &\iff& f|_Q(x) \leq g|_Q(x),\ f|_R(y) \leq g|_R(y) \ \ (\forall x \in Q, y \in R) \\ &\iff& f|_Q \leq g|_Q, f|_R \leq g|_R \\ &\iff& \varphi(f) \leq \varphi(g) \end{eqnarray}
This completes the proof.