And the GPS input to the ADIRS actually comes from the MMRs, so it's a separate component.
We all use FMC position which is derived from several inputs, the IRSs included. What happens to those inputs to produce the position is obviously of some interest to us, (the Aibus FCOMs are very good on this) but of little relevance to our everyday operations. If the ANP is within the RNP, then all is well and generally we are happy. I've used four different types of FMC in my career and all derived their position in different ways. One used a component called an EGI (Embedded GPS IRS) as the main source.