Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations

Assume $\mathsf{Z}\mathsf{F}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{O}\mathrm{N}$ and $⟨E_{\alpha }:\alpha <\kappa ⟩$ be a sequence of equivalence relations on $\mathbb{R}$ with all classes countable and for all $\alpha < \kappa$,$$\mathbb{R}$ / E_\alpha \approx $\mathbb{R}$$\mathrm{.}Thenthedisjointunion$\bigsqcup_{\alpha < \kappa} $\mathbb{R}$ / E_\alpha$isinbijectionwith$$\mathbb{R}$ \times \kappa$and$\bigsqcup_{\alpha < \kappa} $\mathbb{R}$ / E_\alpha$ has the J'onsson property. Assume $\mathsf{Z}\mathsf{F}+\mathsf{A}{\mathsf{D}}^{+}+\mathsf{V}=\mathsf{L}(\mathcal{P}(\mathbb{R}))$. A set $X \subseteq [\omega_1]^{<\omega_1}$hasasequence$\langle E_\alpha : \alpha < \omega_1\rangle$ofequivalencerelationson$$\mathbb{R}$$suchthat$$\mathbb{R}$ / E_\alpha \approx $\mathbb{R}$$and$X \approx \bigsqcup_{\alpha < \omega_1} $\mathbb{R}$ / E_\alpha$ifandonlyif$$\mathbb{R}$ \sqcup \omega_1$injectsinto$X$. Assume $\mathsf{A}\mathsf{D}$. Suppose $R \subseteq [\omega_1]^\omega \times $\mathbb{R}$$isarelationsuchthatforall$f \in [\omega_1]^\omega$,$R_f = {x \in $\mathbb{R}$ : R(f,x)}$ is nonempty and countable. Then there is an uncountable $X\subseteq {\omega }_1$ and function $\Phi : [X]^\omega \rightarrow $\mathbb{R}$$whichuniformizes$R$on$[X]^\omega$:thatis,forall$f \in [X]^\omega$,$R(f,\Phi(f))$. Under $\mathsf{A}\mathsf{D}$, if $\kappa$ is an ordinal and $\langle E_\alpha : \alpha < \kappa\rangle$isasequenceofequivalencerelationson$$\mathbb{R}$$ with all classes countable, then $[{\omega }_1{]}^{\omega }$ does not inject into ${⨆}_{\alpha <\kappa }\mathbb{R}\mathrm{/}{E}_{\alpha }$.