Łoś Theorem: Let $\phi(x_1,...,x_n)$ be a $\mathcal{L}$-formula with free variables in $\{x_1,...,x_n\}$ and a family of $\mathcal{L}$-structures $\{M_i\}_{i\in{I}}$. Let $\mathcal{M}=\prod_{\mathcal{U}}M_i$ an ultraproduct over some ultrafilter $\mathcal{U}$. Then for all $a_1,...,a_n\in\mathcal{M}$ holds
$\mathcal{M}\models \phi[a_1,...,a_n]$ if and only if $\{i\in{I}|M_i\models\phi[{a_1}_i,...,{a_n}_i]\}\in\mathcal{U}$.
Trying to prove it by induction over logical complexity of $\phi$ I find some trouble with the first (and supposedly trivial) case of an atomic formula $\phi(x_1,...,x_n)\equiv p(t_1(x_1,...,x_n),...,t_m(x_1,..,x_n))$. We have
$\mathcal{M}\models\phi[a_1,...,a_n]\iff(t_1^\mathcal{M}(a_1,...,a_n),...,t_m^\mathcal{M}(a_1,...,a_n))\in p^\mathcal{M}\iff\{i\in I|((t_1^\mathcal{M}(a_1,...,a_n))_i,...,(t_m^\mathcal{M}(a_1,...,a_n))_i)\in p^{M_i}\}\in\mathcal{U}$
Where take the $i$-th component of $t_j^\mathcal{M}(a_1,...,a_n)$ make sense because of well definition of the interpretation in the structure $\mathcal{M}$. Now I'd like to continue saying "if and only if $\{i\in I|(t_1^{M_{i}}({a_1}_i,...,{a_n}_i),...,t_m^{M_{i}}({a_1}_i,...,{a_n}_i))\in p^{M_i}\}\in\mathcal{U}$" to conclude the statement for the atomic formula, because this happens if and only if $\{i\in I|M_i\models\phi[{a_1}_i,...,{a_n}_i]\}\in\mathcal{U}.$
Probably it is just a silly misunderstanding of definition of ultraproduct and interpretation of terms in it, but I can't understand how to clearly prove the equivalence in bold, namely that $(t_j^\mathcal{M}(a_1,...,a_n)_i)_{i\in I}$ and $(t_j^{M_i}({a_1}_i,..,{a_n}_i))_{i\in I}$ are always in the same equivalence class in the ultraproduct.
Thanks in advance