@Nigeyy is the bringer of fake 'bad news'.
@Dogtrousers has offered the Sheldon page (scroll almost to the end for the 'width' stuff). As well as the chart, pertinent quote is:
"A general guideline is that the tire width should be between 1.45/2.0 x the inner rim width."
Schwalbe has this:
https://www.schwalbe.com/en-GB/reifenmasse.html
Personally I'd stick to 2" tyres on a 'mountain bike' with slicks if it'll be staying on hard surfaces (ie not going over terrain where tread would enhance grip). And the current rim will be fine from 1.5" to 2.25". The upper limit will likely be dictated by clearance (as others have said).
