Least Fixed Point of Enumeration Operator/Lemma

Lemma
Let $\psi : \powerset \N \to \powerset \N$ be an enumeration operator.

Let $A_i$ be defined recursively as:
 * $A_0 = \empty$
 * $A_{n + 1} = \map \psi {A_n}$

Then $A_i$ is an increasing sequence with respect to the subset ordering.

Proof
By definition of enumeration operator, let $\phi \subseteq \N$ be some recursively enumerable set that yields $\psi$.

Define:
 * $\map P i \iff \forall j < i: A_j \subseteq A_i$

Proceed by induction on $i$.

Basis for the Induction
As there are no natural numbers $j < 0$, $\map P 0$ is satisfied vacuously.

Induction Hypothesis
Suppose the induction hypothesis holds.

That is, suppose that, for $k \in \N$:
 * $\map P k$

or, equivalently:
 * $\forall j < k: A_j \subseteq A_k$

Induction Step
Let $j \le k$ be arbitrary.

By the induction hypothesis, as well as Set is Subset of Itself:
 * $A_j \subseteq A_k$

Therefore, by Subset Relation is Transitive:
 * $A_k \subseteq A_{k + 1} \implies A_j \subseteq A_{k + 1}$

Thus, it suffices to show that $A_k \subseteq A_{k + 1}$.

Suppose $k = 0$.

Then, by definition, $A_0 = \empty$.

Then, $A_0 \subseteq A_1$ follows from Empty Set is Subset of All Sets.

Otherwise, let $k > 0$.

Let $x \in A_k$ be arbitrary.

By definition of enumeration operator, there is a finite subset $B \subseteq A_{k - 1}$ such that:
 * $\map \pi {x, b} \in \phi$

where $b$ is the finite set coding of $B$.

But, by the induction hypothesis, $A_{k - 1} \subseteq A_k$.

Therefore, by Subset Relation is Transitive:
 * $B \subseteq A_k$

Additionally, as $B$ is finite by definition, it follows that $B$ is a finite subset of $A_k$.

As $\map \pi {x, b} \in \phi$, it follows that $x \in A_{k + 1}$ by definition.

As $x$ was arbitrary, it follows from the definition that:
 * $A_k \subseteq A_{k + 1}$

As the Basis for the Induction holds, and the Induction Step follows from the Induction Hypothesis, we conclude by the Principle of Mathematical Induction that:
 * $\forall i \in \N, j < i: A_j \subseteq A_i$

Therefore, the sequence $A_i$ is increasing with respect to the subset ordering by definition.