x + y² + z³ = 3
y + z² + x³ = 3
z + x² + y³ = 3

Can you prove that it is the only such solution?

With the substitution x = 1+d_x, y=1+d_y, z=1+d_z, the domain x>0, y>0, z>0 gets remapped to d_x>-1, d_y>-1, d_z>-1.
Your proof only goes as far as d_x>0, d_y>0, d_z>0, which only covers x>1, y>1, z>1, and does not rule out possibilities like 1>x>0.