My textbook of physics, Gettys's Physics (Italian language edition), says that the Lorenz gauge choice uses the magnetic vector potential $$\mathbf{A}(\mathbf{x},t):=\frac{\mu_0}{4\pi}\int_{\mathbb{R}^3} \frac{\mathbf{J}(\mathbf{y},t-c^{-1}\|\mathbf{x}-\mathbf{y}\|)}{\|\mathbf{x}-\mathbf{y}\|}d\mu_{\mathbf{y}} $$ where $\mathbf{J}:\mathbb{R}^4\to\mathbb{R}^3$ is a function satisfying the assumptions given in physics, like differentiability and being zero outside $R\times\mathbb{R}$ where $R\subset\mathbb{R}^3$ is a bounded domain, $\mu_0$ is magnetic permeability and $c$ is the speed of light.
How can we prove that such an $\mathbf{A}$ satisfies Maxwell's equation $$\nabla\cdot(\nabla\times\mathbf{A})=0?$$ Problems arise since I am not sure whether and how we could differentiate under the integral sign, and I am asking here rather than in PSE because I notice that questions focussing on mathematical derivations tend to be redirected here.