The priesthood of Christ is derived from sacrifice

"In what was done by Melchisedech the priest we recognize a type of the Sacrament of the Lord's Sacrifice.  For thus it is written in the writings of God: And Melchisedech, King of Salem, brought forth bread and wine, for he was the priest of the Most High God, and he blessed Abraham.  Concerning the fact that Melchisedech was a type of Christ, the Holy Ghost himself doth testify in the Psalms, where the First Person of the Holy Trinity (that is, the Father) is set before us as saying unto the Second Person (that is, the Son): Before the day-star have I begotten thee: Thou art a priest for ever, after the order of Melchisedech.  And doubtless the sameness of order in the priesthood of Christ and of Melchisedech is derived from sacrifice, and proceedeth from this, namely; that Melchisedech was the priest of the Most High God; that he offered bread and wine; and that he blessed Abraham.

For who is so truly the priest of the Most High God as is our Lord Jesus Christ?  And he it is that hath made an offering unto God the Father, and the same offering that Melchisedech made, Bread and Wine, that is to say, his own Flesh and his own Blood.  And so far as Abraham is concerned, the blessing which Melchisedech gave him so long ago belongeth also to us.  For if Abraham believed God and it was accounted to him for righteousness, verily then, whosoever believeth God and liveth by faith, the same is found righteous, and is made manifest unto us as one who hath thereby attained the blessing given faithful Abraham; which same is also justified as the Apostle Paul proveth, where he saith: Abraham believed God, and it was accounted to him for righteousness; know ye therefore that they which are of faith, the same are the children of Abraham; and the Scripture, foreseeing that God would justify the heathen through faith, preached before the Gospel unto Abraham, saying, In thee shall all nations be blessed.

To the end therefore, that this blessing of Abraham by Melchisedech the priest might be duly solemnized, it was preceded (as we are told in Genesis) by a symbolic sacrifice consisting of bread and wine.  Completing and fulfilling this sacrifice, our Lord Jesus Christ offered up bread, and a cup of wine mingled with water.  And thus he who came, (not to destroy, but to fulfil, the Law and the Prophets,) utterly satisfied all the implications prefigured in the oblation made by Melchisedech.  Through Solomon's Proverbs also did the Holy Ghost clearly foreshadow, as it were in a parable, the Lord's Sacrifice, saying: Wisdom hath builded her house, she hath hewn out seven pillars: referring thus to the Church.  In the same passage he pointeth to the victim slain, and the bread and wine, saying: She hath killed her beasts, she hath mingled her wine.  He pointeth to the altar in the words: She hath also furnished her table.  And to the apostolic priesthood in the words: She hath sent forth her servants, she crieth upon the highest places of the city, saying, Whoso is simple, let him turn in hither unto me; as for them that want understanding, she saith to them, Come, eat of my bread, and drink of the wine which I have mingled for you."


-- From the letter to Caecilius by St. Cyprian, Bishop and Martyr