Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

Question

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ implies that $f(x) \in V$)}$$

Answer 1

See Restricted quantifiers:

$(\forall x \in D)P(x)$ is equivalent to: $\forall x (x \in D \to P(x))$.

Thus, your formula: $∀x \in U : f(x) \in V$ is equivalent to:

$\forall x (x \in U \to f(x) \in V)$.