User:Dfeuer/Progressing Function Lemma

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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.

$\blacksquare$