Prove that two functions f and g are equal if and only if...

Prove that two functions f and g are equal if and only if

1. The domain of f is equal to the domain of g

and

2. for all x that are elements of the domain of f, f(x)=g(x)

The domain simply refers to the first coordinate for all coordinates in the function.

Proof:
Assume f=g

1. Suppose that x is an element in the domain of f, then (x,y) is an element of f for some y. Since we assume that f=g then (x,y) is also an element of g and x must be an element in the domain of g. Thus the domain of f must be a subset of the domain of g, and similarly g must be a subset of f. So the two domains are equal.

2. Again suppose that x is an element of the domain of f. Then for some y, (x,y) is an element of f. Since f=g, (x,y) must also be an element of g. Therefore f(x)=y=g(x)