In math, as opposed to in computer science, when you apply "mod y" you land in the integers modulo y, denoted Z/yZ, not back in the integers. This means that mod 7, the symbols 1 and 8 denote the same thing, i.e. the equivalence class {...,-13,-6,1,8,15,...}.
A more computer-y way to say this is that for mathematicians, "integers integer mod 7" is a different kind of data class than "integer."
All this is just a long way of saying that this is probably the wrong place to ask your question.
If I were defining this function in a math paper I'd say something like "Let f(x,y) denote the unique number in {1,2,...,y} which is congruent to x modulo y" or "By the division algorithm there exists a unique number in {1,2,...,y} which is congruent to x modulo y, we denote this number f(x,y)."
