Reflexive Closure is Inflationary

Theorem
Let $S$ be a set.

Let $R$ denote the set of all endorelations on $S$.

Then the reflexive closure operator is an inflationary mapping on $R$.

Proof
Let $\mathcal R \in R$.

From Reflexive Closure/Union with Diagonal we define $\mathcal R^=$, the reflexive closure of $\mathcal R$, to be:


 * $\mathcal R^= := \mathcal R \cup \Delta_S$

From Subset of Union/Binary Union:


 * $\mathcal R \subseteq \mathcal R^=$

Hence the reflexive closure operator is an inflationary mapping.