I am proving a kind of structure invariance principle for magmas in Cubical Type Theory with the Agda/Cubical library. This is done by constructing a path between two simple magmas and then transporting proofs of simple properties about this path. I have already obtained most of the proof (see my code repository) but did not manage yet to complete the following lemma.

At some point in the proof I have the following given:

- A bijection between types:
`f : ℕ → ℕ₀`

- The isomorphism constructed with
`f`

:`fIso : Iso ℕ ℕ₀`

- A function that gives the inverse isomorphism of an isomorphism:
`invIso : Iso A B → Iso B A`

Now, I would like to prove that:

`sym (ua (isoToEquiv fIso)) ≡ ua (isoToEquiv (invIso fIso)) `

There are two parts to my question:

- Is this a valid theorem in HoTT? Although this statement seems valid, maybe I have produced a false statement?
- Are there built-in functions in the Agda/Cubical that may help in the proof?