Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 6 / iii / b / Solution

Equality with the union is expressed byThe first conjunct says that every member of $x$ lies in some member of $y$; the second gives the reverse inclusion. Every quantifier is bounded by a parameter or by an element of a previously bounded parameter. Therefore bounded absoluteness applies, proving full absoluteness and hence upward absoluteness of $x=\bigcup y$. This only compares the relation for existing parameters; it does not require either class to contain the union of every one of its sets.