On my irish residence card not mentioned that I’m family member of eu National.

I’m on my way to spain from uk with my wife who is eu National and I’m her spouse.i have irish stamp 4.

But the airline authorities did not allowd to me because on my residence card did not mentioned my wife’s name.

We just don’t understand how it could be because we are flying from Dublin to Birmingham without any problem.
I have all my document my pasport,stamp 4,and are marriage certificate.

Do you have stamp 4 or 4EUFAM? No residence card would have a spouse’s name on it. You must have confused it with something else.
If you have Stamp 4, you are rightly denied boarding by the airline.
Travel inside common travel area is much simpler. Although certain airlines may still want to check the residence eligibility, most won’t be bothered.