The phrase "firmly established" implies a sense of stability, longevity and reliability. Synonyms for this phrase include the words "well-established", which is used to describe something that has been in place for a considerable amount of time, and has gained a reputation for being dependable and trustworthy. Another synonym that comes to mind is "rooted", which implies a deep sense of permanence and stability, as though the subject is firmly anchored in place. Other options include "long-standing", which again suggests that something has been in place for a considerable amount of time, or "settled", which speaks to the idea that something is fixed and unchanging.