Definition of a solution of a first order ODE

jacksonjs20

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

[itex]\textbf{Question}[/itex]: 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.