Consistency of ZFC + "for every function there exists a class inaccessible to it"

Question

Is ZFC + the following statement consistent (and if so, is it equiconsistent to some known large cardinal):

For every function $f:ORD \rightarrow ORD$ such that:

1. $f(\alpha)\geq \alpha$,

2. $\alpha > \beta \Rightarrow f(\alpha) \geq f(\beta)$

$\{\kappa|\kappa \text{ is regular} \wedge \forall \alpha <\kappa (f(\alpha)<\kappa)\}$ is a proper class.

This is obviously not a theorem of ZFC, as we can take $f(\alpha)=|2^{\aleph_\alpha}|$ and conclude that we have a proper class of inaccessible cardinals. Actually, using $f(\alpha) = \alpha\text{-th-inaccessible cardinal}$ we get 1-inaccessible cardinals, and then we can get hyper-inaccessibles and get at least as far up as Mahlo cardinals, maybe more.

So either this statement is somehow plainly inconsistent, or it has a consistency strength of some large cardinal.

Note: the idea behind this construction is to "take away the uniqueness" of the power function in constructing the inaccessibles. Literally, what's 'the most' we can get out of a particular function.

Answer 1

This is also known as "$\rm Ord$ is Mahlo". Namely, every class which is closed and unbounded has a regular cardinal in it.

It is certainly weaker than a Mahlo cardinal: If $\kappa$ is Mahlo, then $\langle V_\kappa,\in\rangle$ is a model of this theory. But this is weaker. We only need every definable class to have a regular cardinal, and in $V_\kappa$ every club will have a cardinal.

So we can reduce it a bit from Mahlo to some sufficiently inaccessible cardinal. See also http://cantorsattic.info/ORD_is_Mahlo for some details.