Question about equivalence relation that concludes C=D

Question

Here is a statement I retrieved from Munkres Topology. It states that for x in a set A

yCx if and only if yDx concludes that C=D. I don’t quite understand how this conclusion come from.

Answer 1

From the context, I am guessing that $C$ and $D$ are relations on the same set $A$. The formal definition of relation is a set of ordered pairs. So both $C$ and $D$ are subsets of $A \times A$.

The syntactic sugar notation for a relation $R\subseteq A \times A$ is to write $x \mathrel{R} y$ when $(x,y) \in R$. So the statement $y \mathrel{C} x \iff y \mathrel{D} x$ is syntactically equivalent to $(y,x) \in C \iff (y,x) \in D$. But this is equivalent to $C = D$ as sets.