Characterization of Strictly Increasing Mapping on Woset

Lemma
Let $J$ and $E$ be well-ordered sets.

Let $h: J \to E$ be a mapping.

Let $S_\alpha$ denote an initial segment determined by $\alpha$.


 * $(1):\quad$ $h$ is order-preserving and its image is either all of $E$ or an initial segment of $E$


 * $(2):\quad$ $\forall \alpha \in J: h\left({\alpha}\right) = \min \left({E\setminus h\left[{S_\alpha}\right]}\right)$

where:


 * $h\left[{S_\alpha}\right]$ denotes the image of $S_\alpha$ under $h$


 * $\min$ denotes the smallest element of the set.

$(1)$ implies $(2)$
Suppose $h$ satisfies:


 * $h$ is order-preserving and its image is either all of $E$ or an initial segment of $E$

Then for any $x,y \in J$:

$(2)$ implies $(1)$
Suppose $h$ satisfies:


 * $h(\alpha) = \min\left({E \setminus h\left[{S_\alpha}\right] }\right)$

By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined.

Then:

Thus for every $x \in S_y$, we have that $h(x) \in S_{h(y)}$.

Therefore $h$ is an order-preserving mapping.

Furthermore, the image set of $h$ is the union of initial segments in $E$.

By Union of Initial Segments is Initial Segment or All of Woset, $h[J]$ is an initial segment of $E$ or all of $E$.