Proposition 5.4. Assuming that:

${(A_{j})}_{j \in J}$ , ${(B_{j})}_{j \in J}$ satisfy $B_{j} \subseteq A_{j}$ for all $j$
${(a_{j})}_{j \in J}, {(b_{j})}_{j \in J} \in \prod_{j \in J} A_{j}$ satisfying ${(a_{j})}_{j \in J} \sim {(b_{j})}_{j \in J}$

Then

{j \in J : a_{j} \in B_{j}} \in U

if and only if

{j \in J : b_{j} \in B_{j}} \in U