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.

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 a 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.