Property of Increasing Mapping on Ordinals

Theorem

Let $\On$ denote the class of all ordinals.

Let $K_{II}$ denote the class of all limit ordinals.

Let $F$ be a mapping defined on $\On$ satisfying the following conditions:

$\text {(1)}: \quad$

$\ds \forall \alpha \in \On: \, $

$\ds \map F \alpha$

$\subseteq$

$\ds \map F {\alpha^+}$

$\text {(2)}: \quad$

$\ds \forall \lambda \in K_{II}: \forall \alpha \in \On: \, $

$\ds \alpha$

$<$

$\ds \lambda$

$\, \ds \implies \, $

$\ds \map F \alpha$

$\le$

$\ds \map F \lambda$

Then:

$\ds \forall \alpha, \beta \in \On: \, $

$\ds \alpha$

$\le$

$\ds \beta$

$\, \ds \implies \, $

$\ds \map F \alpha$

$\le$

$\ds \map F \beta$

Proof

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Sources

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Exercise $5.4$