User:Dfeuer/Progressing Function Lemma

Theorem
Let $g$ be a $\subseteq$-inflationary mapping.

Define a relation $R$ on $\operatorname{dom} g \times \mathbb U$ by letting $xRy \iff (g(x) \subseteq y) \lor (y \subseteq x)$.

Then for all $x$ and $y$ in $\operatorname{dom}g$:


 * $(1)\quad x R \varnothing$
 * $(2)\quad x R y \land y R x \implies x R g(y)$

Proof
This follows from User:Dfeuer/Progressing Function Lemma/General Result applied to the class $\mathbb U$ with the subset ordering.