Miscellaneous Facts about Functions

For simplicity we adopt the following rules: x is arbitrary, m, n are natural numbers, f , g are functions, and A, B are sets. We now state several propositions: (1) For every function f and for every set X such that rng f ⊆ X holds idX · f = f. (2) Let X be a set, and let Y be a non empty set, and let f be a function from X into Y . Suppose f is one-to-one. Let B be a subset of X and let C be a subset of Y . If C ⊆ f B, then f −1 C ⊆ B. (3) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let x be an element of X and let A be a subset of X. If f(x) ∈ f A, then x ∈ A. (4) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let x be an element of X, and let A be a subset of X, and let B be a subset of Y . If f(x) ∈ f A \B, then x ∈ A \ f −1 B. (5) Let X, Y be non empty sets and let f be a function from X into Y . Suppose f is one-to-one. Let y be an element of Y , and let A be a subset of X, and let B be a subset of Y . If y ∈ f A\B, then f−1(y) ∈ A\f −1B. (6) For every function f and for arbitrary a such that a ∈ dom f holds f {a} = a7−→ . f(a).