Concretely, $\smash {\mathord {\sim }^{\mathrm{eq}}_{R}}$ is the equivalence relation on $A$ defined by
\begin{align*} R^{\mathrm{eq}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\webleft (R^{\mathrm{refl}}\webright )^{\mathrm{symm}}\webright )^{\mathrm{trans}}\\ & = \webleft (\webleft (R^{\mathrm{symm}}\webright )^{\mathrm{trans}}\webright )^{\mathrm{refl}}\\ & = \webleft\{ \webleft (a,b\webright )\in A\times B\ \middle |\ \begin{aligned} & \text{there exists $\webleft (x_{1},\ldots ,x_{n}\webright )\in R^{\times n}$ satisfying at}\\[-2.5pt]& \text{least one of the following conditions:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{The following conditions are satisfied:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $a\sim _{R}x_{1}$ or $x_{1}\sim _{R}a$;}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $x_{i}\sim _{R}x_{i+1}$ or $x_{i+1}\sim _{R}x_{i}$}\\[-2.5pt]& \mspace {81.25mu}\text{for each $1\leq i\leq n-1$;}\\ & \mspace {50mu}\rlap {\text{(c)}}\mspace {30mu}\text{We have $b\sim _{R}x_{n}$ or $x_{n}\sim _{R}b$;}\\[7.5pt]& \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{We have $a=b$.}\end{aligned} \webright\} .\end{align*}
From the universal properties of the reflexive, symmetric, and transitive closures of a relation (Definition 8.1.2.1.1, Definition 8.2.2.1.1, and Definition 8.3.2.1.1), we see that it suffices to prove that:
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The symmetric closure of a reflexive relation is still reflexive;
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The transitive closure of a symmetric relation is still symmetric;
which are both clear.
