Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element/Proof

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Theorem

Let $N$ be a class which is closed under a progressing mapping $g$.

Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$.


$g$ has no fixed point, unless possibly the greatest element, if there is one.


Proof

Suppose $g$ has a fixed point.

Let $x$ be an element of $M$ such that $\map g x = x$.


We have from Smallest Element of Minimally Closed Class under Progressing Mapping that:

$b \subseteq x$

Suppose that $y \subseteq x$.

Then by Image of Proper Subset under Progressing Mapping on Minimally Closed Class:

$\map g y \subseteq \map g x$

But we have that $\map g x = x$.

Thus:

$\map g y \subseteq x$


So we have:

$b \subseteq x$

and:

$y \subseteq x \implies \map g y \subseteq x$

and the result follows by the Principle of General Induction for Minimally Closed Class.

$\blacksquare$


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