A question about a property of independent random variables

Question

Let $(\Omega,\mathcal F, P)$ be a probability space, and $X:\Omega \to \mathbb R$ and $Y:\Omega \to \mathbb R$ be two independent random variables. Let $f:\mathbb R^2 \to \mathbb R$.

I want to show that if $$ \forall z \in \mathbb R, \Pr(f(z,Y(\omega))<0)\geq 1-\delta, $$ then $$ \Pr(f(X(\omega),Y(\omega)) < 0) \geq 1-\delta. $$ How do I show this?