User blog comment:Deedlit11/Ordinal notations II: Up to the Bachmann-Howard ordinal/@comment-11227630-20170727144618

I've seen some ordinal collapsing functions defined in a $$\vartheta$$-way, but the condition "$$C(\alpha,\beta)\cap\Omega\subseteq\beta$$" is modified to "$$C(\alpha,\beta)\cap\Omega=\beta$$". What's the difference between the "$$\subseteq$$ version" and the "$$=$$ version"? What about higher collapsing functions (such as this one)?