"In this sentence I think yo..."

If $f$ maps $X$ to a subset $I$ of $Y$, why not just say that its domain is $I$ and do away with the distinction between domain and codomain? There are at least two cases where the distinction is useful\.

In this sentence I think you meant 'codomain' where you said 'domain' and 'image' where you said 'codomain'. Is that right?