We give a new notion of displayed setoid (family of setoids) in intensional type theory. It is used to give a semantics of extensional type theory with universes (ETU) within Agda with
options --safe and --without-K, used as a machine-checkable formalization of intentional type theory augmented with a universe closed under inductive-recursive definitions (ITU). The syntax of ETU is defined in safe Agda in a traditional extrinsic form, using an Agda library WSLN for well-scoped locally nameless representation of its expressions. Giving its semantics in the ITU setoid model is complicated by the very limited means of expression afforded by ITU. As a corollary we obtain a proof of the consistency of ETU within ITU.
Checked with Agda version 2.8.0 using options
--safe
--without-K
--no-postfix-projections