By definition ={ x:xεX & f(x)εB }.....................................1
So we see that IS A subset of X. Now since for any subset ,lets say A, of X we have by definition f(A) ={ f(x) : xεA } then ={ f(x): xε }................................................. ...........2
So we see that is a set of the images of x f(x) , such that xε .
The question now is are these f(x)'s inside B??
Lets see:
Let .........................f(x)ε then from 2 we infer f(x)=f(x) & xε .
But from 1 if xε we conclude xεX & f(x)εB.
Hence f(x) is inside B.AND is a subset of B
Now for the converse let f(x)εB. Since f is onto there exists an xεX such that f(x)εB AND so by 1 xε ........................e.t.c,..........e.t.c