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.
$\blacksquare$