In this video Dr Woodhead discusses Adam’s first exercise of his new dominion since the Fall is to provide a name for his wife whom he previously only called “The Woman” (Genesis 2: 23). Adam stated that the reason he called her woman was because she was taken out of man.
In Hebrew and in English there is a wordplay between the words “man” (אִישׁ, ˒îšh) and “woman” (אִשָּׁה, ˒išhāh)
The Hebrew words for man and woman are ish and isha. It amounts to a sort of play-on-words. Man was not made from women initially; Woman was made from man. Now in his post Fall state he calls her Eve. The Hebrew word for Eve is Chavah, which means life. He calls her this because she would be the mother of all living humans on earth. This affirms that even though their sin produced death and decay her name signified that life would go on. They would procreate and fill the earth. That commandment to fill the earth had not been nullified with the Fall.