Definition 1.5.9.  Let X be a set, A ⊆ X, Y be a set, B ⊆ Y, f:X → Y be a function such that f(A) ⊆ B. We define:

This definition is a slight extension of $restriction that simultaneously restricts the codomain to a subset B of Y, under the condition that the range is contained in this subset. It is compatible with the previous definition.