Cartesian Product of Functions

For simplicity we follow the rules: x, y, y1, y2, z, a will be arbitrary, f , g, h, h′, f1, f2 will denote functions, i will denote a natural number, X, Y , Z, V1, V2 will denote sets, P will denote a permutation of X, D, D1, D2, D3 will denote non-empty sets, d1 will denote an element of D1, d2 will denote an element of D2, and d3 will denote an element of D3. We now state a number of propositions: (1) x ∈ ∏ 〈X〉 if and only if there exists y such that y ∈ X and x = 〈y〉. (2) z ∈ ∏ 〈X,Y 〉 if and only if there exist x, y such that x ∈ X and y ∈ Y and z = 〈x, y〉. (3) a ∈ ∏ 〈X,Y,Z〉 if and only if there exist x, y, z such that x ∈ X and y ∈ Y and z ∈ Z and a = 〈x, y, z〉. (4) ∏ 〈D〉 = D1. (5) ∏ 〈D1,D2〉 = {〈d1, d2〉}. (6) ∏ 〈D,D〉 = D2.