Proof involving surjective/onto and image/preimage of sets

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EXERCISE: Suppose f is surjective, and B is a subset of Y. Prove that f(f^-1(B))=B.

SOLUTION: We must show that f(f^-1(B)) is a subset of B and that B is a subset of f(f^-1(B)). I have already proven that f(f^-1(B)) is a subset of B. Now I must prove that B is a subset of f(f^-1(B)) when f is surjective. Fix x is an element of B.

After this I am lost. Help please! I know that the surjectivity must come in handy at some point since B is not a subset of f(f^-1(B)) for all f.
 
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Well

f^{-1} : B \rightarrow Y

So let x be in B as you said. Now f^{-1}(x) is an element of Y provided that there is some y in Y such that f^{-1}(x) = y. This is where you need to use surjectivity. Hope this helps.
 

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