'' mixed set of French and English institutions that existed in the Province of Quebec during the 1763–91 period and which continued to exist later in Canada.''
Not exactly. Bas-Canada is basically what is now the province of Québec, earlier was Canada, Nouvelle-France, which was the French colony in the St-Lawrence valley. There were other parts of New France, such as the ''Pays d'en Haut'' (now Ontario and various US territories such as Détroit etc), the ''Pays des Illinois'' and ''Louisiane''. These last two are both now part of the USA, what they consisted of was pretty broad.
The institutions that existed during New France days were operating under the French laws of the time. When the English conquest happened, there was a fair bit of wrangling as to which institutions would change and which would stay the same. What we finally wound up with was that civil law was based on the old French system, obviously altered over time, and to this day Québec's civil law is different than that of the other provinces. Criminal law was established under the English system after the conquest, which is probably why to this day it is under federal jurisdiction.