Let $N$ be a nonempty subset of the universe $M$ of a $L$-structure $\mathcal{M}$.
Show: $N$ is the universe of a substructure $\mathcal{N}$ if and only if it is closed under $t^\mathcal{M}$ for all $L$-terms $t=t(x_1, \cdots , x_n)$
Attempt:
$(\Rightarrow)$
Suppose $N$ is the universe of a substructure $\mathcal{N}$. Then by definition we have
i) $c^{\mathcal{N}}=c^{\mathcal{M}}$ for all constants $c \in L^{\mathcal{N}}$.
ii)$f^{\mathcal{M}} \restriction_{\mathcal{N}^n}= f^{\mathcal{N}}$.
Since every constant symbol is and function symbol is a $L$-term, then by i) ii) we are done.
$(\Leftarrow)$
Suppose $N$ is closed under $t^\mathcal{M}$ for all $L$-terms $t=t(x_1, \cdots , x_n)$. Then i) ii) hold?
I'm not sure how to expand the proof from i) ii). More specifically, I don't really under stand it means to be"closed under $t^\mathcal{M}$ for all $L$-terms $t=t(x_1, \cdots , x_n)$".
Any ideas or hints will be appreciated!
Thanks!