Complex sentence: Formally, an indexed family is the same thing as a mathematical function; a function with domain "J" and codomain "X" is equivalent to a family of elements of "X" indexed by elements of "J".
Split into simpler sentences: Formally, an indexed family is the same thing as a function. A function with domain "J" and codomain "X" is equivalent to a family of elements of "X" indexed by elements of "J".