The following appears as fact 3.1 in the slides from Hjorth's 2010 Tarski Lectures. Assume ${L(\mathbb R)} \models \mathrm{AD}$.

Fact 3.1 For E and F Borel equivalence relations one has $$E \leq_{L(\mathbb R)} F$$ if and only if $$|\mathbb R/E|_{L(\mathbb R)} \leq |\mathbb R /F|_{L(\mathbb R)}.$$

My understanding is that the first inequality is a reduction via a function in ${L(\mathbb R)}$, and the second inequality is comparison of cardinalities within ${L(\mathbb R)}$. What is the proof of this result? I am particularly interested in whether this result holds in the Chang model, assuming convenient large cardinal axioms.