The word, "Holy" comes from the Hebrew word, qodesh (Strong's H6944), which means apartness, set-apartness, of God, of places, of things, also, set-apartness, separateness, sacredness. So the Holy Spirit is indeed of God, especially when viewed in His separate operative capacity, like "comforter", "teacher", one by whom we are "sealed", etc.
This is not unlike the separate operative capacity of the WORD of God--Spirit--who was made flesh to be judged for the sin of the world by being separated from God the Father in Hell for three days and three nights as prophesied in Psalm 22, And, as fulfilled in
And about the ninth hour Jesus cried with a loud voice, saying, Eli, Eli, lama sabachthani? that is to say, My God, my God, why hast thou forsaken me?
and finishing with the same cry, in verse 50:
Jesus, when he had cried again with a loud voice, yielded up the ghost. (My emphasis)
And, in Acts 2:25-27:
For David speaketh concerning him, I foresaw the Lord always before my face, for he is on my right hand, that I should not be moved: Therefore did my heart rejoice, and my tongue was glad; moreover also my flesh shall rest in hope: Because thou wilt not leave my soul in hell, neither wilt thou suffer thine Holy One to see corruption. (My emphasis)
David was consoled concerning his own soul because of his faith in the promise that God would not suffer His Holy One to see corruption, neither would He leave David's own soul in Hell. As it was written, in Psalm 16:10:
For thou wilt not leave my soul in hell; neither wilt thou suffer thine Holy One to see corruption. (My emphasis)
And again in Matthew 12:40:
For as Jonas was three days and three nights in the whale's belly; so shall the Son of man be three days and three nights in the heart of the earth.
The heart of the earth was not in that tomb where Jesus' body remained, but rather, in the heart of the earth--in hell--in Abraham's bosom--separated by a great gulf fixed from unbelievers (read Luke, chapter 16 in total). Those believers in Abraham's bosom in hell at that time, although separated as being believers from unbelievers, were not in heaven. They, too, were in the heart of the earth separated from their own long-ago decayed bodies, because Jesus had not yet completed the necessary work for their salvation. Indeed, He was now separated from the Father, just like them, but now positioned to preach to ALL those spirits in prison words that had they been waiting years, decades, and even thousands of years to hear--that God's Holy One had finally died as an atonement for their sin, as it is written in 1 Peter 3:18-19:
For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit: By which (Spirit) also he went and preached unto the spirits in prison; (My insertion for clarification)
That Word of God revealed the separate operational capacities of Himself, the Speaker, the Lord God, and the His (the Lord God's) Spirit, the Senders, in Isaiah 48:16:
Come ye near unto me, hear ye this; I have not spoken in secret from the beginning; from the time that it was, there am I: and now the Lord GOD, and his Spirit, hath sent me. (My emphasis)
These three separate operational capacities of the ONE TRUE GOD indeed have individual functions and they always stayed in their own lane, respectively, while accomplishing ETERNAL SALVATION for everyone who believes in the separate, but dovetailing accomplishments of the Father, the Son, and the Holy Ghost. Those accomplishments of the Holy Father, Holy Son, and Holy Ghost are brilliantly highlighted as basic necessities for teaching by the great commission given to the apostles, who were likewise "sent" to "teach" all nations by baptizing (submerging) them in the name of the three individual operatives of the ONE True God, as we see in Matthew 28:19-20:
Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you alway, even unto the end of the world. Amen. (My emphasis)
So the Holy Spirit is indeed God, Even as the WORD who was with God and who was God, is God, and as the Lord God is God--the ONE TRUE GOD--the "us" in Genesis 1:26:
And God said, Let us make man in our image, after our likeness:.... (My emphasis)
So, yes, the Holy Spirit is of the Father.