If $\forall y\in B, y=h(x)$ where $x\in A$, is it true that $\exists y$ s.t. $y=f(x), x\in A$ for every $y\in B$?

Question

I know it is not great grammar, but is it logically incorrect to say something like

If $\forall y\in B, y=h(x)$ where $x\in A$, it is true that $\exists y$ s.t. $y=f(x), x\in A$ for every $y\in B$

Answer 1

If you're saying $[\forall y \in B (f(y) = x)] \implies [\forall y \in B (\exists y \in B (f(y) = x))]$, this is indeed true. If you're saying $[\forall y \in B (f(y) = x)] \implies [\exists y \in B (f(y) = x)]$, this is not necessarily true (for consider $B = \emptyset$).