Firmly established in a person's beliefs or habits; deeply ingrained in the nature of a person or thing
John Major was described by his opponents as a dyed-in-the-wool Conservative.
If you use dyed-in-the-wool to describe someone or their beliefs, you are saying that they have very strong opinions about something, which they refuse to change. a dyed-in-the-wool conservative. having strong beliefs, likes, or opinions that will never change