Let $F:[0,1]\times[0,1]\to \mathbb {R}^2$ be a smooth function. Given $x\in [0,1]$, let $\ell_x:=\{x\}\times [0,1]$, and given $y\in [0,1]$, let $\ell_y:=[0,1]\times \{y\}$.

My question feels intuitive, by I have failed for a long time to find a rigorous argument. The most naive version of it is the follows:

Is there a constant $C>0$ s.t for all closed $A\subseteq [0,1]\times[0,1]$, $$\mathrm{Area}(F[A])\leq C\cdot \max_{x\in[0,1]}\mathrm{Length}(F[\ell_x\cap A])\cdot \max_{y\in[0,1]}\mathrm{Length}(F[\ell_y]) ?$$

Any drawing I make makes me feel like this should be true, but I cannot prove it. Moreover, it is of interest to me to understand what happens if the image of $F$ is in fact a manifold of bounded curvature?

Any help, guidance, or reference would be appreciated!