score:4
Your statement is correct in principle, but the vocabulary could lead to some incorrect conclusions.
the father does not require the son or holy spirit to exist
could be understood to mean that at some time the Father could have existed (or possibly even did exist) without the Son or the Spirit existing. This is not correct as the Son and the Spirit are co-eternal and co-uncreated with the Father.
Likewise,
the son and holy spirit are dependent on the father for their existence
is true as the Father is the source of the Trinity. However we must not understand 'dependent' to mean the Son and the Spirit are in someway conditional -- that if things had gone differently there could have been (for example) a Father, Sister, and Heartbeat instead.
The Father is the Fountainhead of the Holy Trinity, and it is not possible for there to be a Fountainhead without that which flows from it, the Son begotten and the Spirit proceeding.