I have to prove that for my discrete class, and I know it's true, but I'm stuck. I stopped to see my professor but I am still confused. Any help is greatly appreciated.

I started the proof going left to right:

pf: Let x

f(S U T). Then f(x)

S U T and there exists a z

S U T such that f(z) = x

Since z

S U T then z

S or z

T.

Assume z

S. Since f(z) = x, then x

f(S) and x

f(S) U f(T).

Assume z

T. Since f(z) = x, then x

f(T) and x

f(S) U f(T).

Therefore, since x

f(S) U f(T), and f(S U T) is a subset of f(S) U f(T).

I know I need to go back the other way, and as long as the part shown is correct then the other way is too. Any criticism is welcome, or tell me where I went wrong, etc.

Thanks again in advance for any help!!