So I am pretty sure this statement is false so my proof should be incorrect by I cannot see where the error is.

The statement is: If g o f is 1 -1 then g is 1 -1

Let (f(x),g[f(x)]), (f(z),g[f(z)]) be elements of g

to prove 1 -1 assume g[f(x)] = g[f(z)]

so we must prove f(x) = f(z) in order to be 1 - 1

(x, g[f(x)]), (z, g[f(z)]) are elements of g o f

since g[f(x)] = g[f(z)] and g o f is 1 - 1, x = z

(x, f(x) ), (z, f(z) ) are elements of f

since x = z and f is a function, f(x) = f(z)

therefore g is 1 - 1.

Can someone help?