Open Immersions Stable under Base Change
Solution 1:
The lemma states that if you have two $S$-schemes $X'$ and $Y',$ and open subschemes $U\subseteq S,$ $V\subseteq X',$ and $W\subseteq Y',$ the canonical map $$V\times_U W\to X'\times_S Y'$$ is an open immersion.
To obtain your statement from this lemma, Set $X' = U = S,$ $V = X,$ and $W = Y' = Y.$ Then the open immersion in the lemma becomes $$V\times_U W = X\times_S Y\to X'\times_S Y' = S\times_S Y = Y,$$ which is what you wanted to show was an open immersion.