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**billa** $\displaystyle

A\subseteq B$ : given

$\displaystyle

R[A]=\{y\mid(\exists x\in A)(x,y)\in R\}

$

$\displaystyle

R[ B ]=\{y\mid(\exists x\in B)(x,y)\in R\}

$

$\displaystyle (\forall x\in A)\,\, x\in B$

$\displaystyle

(\forall y\in R[A])\,\, y\in\{y\mid(\exists x\in B)\,(x,y)\in R\}

$

$\displaystyle

R[A]\subseteq R[ B ]

$

I have to proof the last line in that list; does this constitute a proof?

Thanks, I am unused to both set theory and proofs, which makes this especially hard.