Definition of a solution of a first order ODE

Given an open connected subset $D$ of the $(t,x)$ plane and a function $f\in C(D,\mathbb{R})$, we say $\varphi\in C^1(\text{proj}_1D,\mathbb{R})$ is a solution of the first order differential equation $x'=f(t,x)$ if and only if $$\forall t\in \text{proj}_1D,\quad (t,\varphi(t))\in D$$ and
$$\forall t\in I, \quad \varphi'(t)=f(t,\varphi(t)) .$$

$\textbf{Question}$: Is there a way to alter this definition so that the first condition after the 'iff' is automatically satisfied?

Thanks in advance for any help.
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