Let $X, Y, Z$ be metric spaces such that $X, Y$ are compact. Let $F: X \times Y \to Z$ be a continuous function. If $x \in X$ we defined $F_x:Y \to Z$:
$$F_x(y)=F(x,y)$$
for every $y \in Y$. I need to prove that $\mathcal{F} = \{F_x \}_{x \in X}$ is bounded and uniformly equicontinuous.
$X \times Y$ is compact then $\vert F \vert$ has a maximum value $M$ then for every $x \in X$, $F_x$ is bounded by $M$. Where I have the problem is proving that is uniformly equicontinuous and what is the difference between uniformly equicontinuous and equicontinuous?