Not so. For example, this basic Halo RL 4 in. White Integrated LED Recessed Ceiling Light Trim emits 915 lumens which it claims is the output of a 60 watt bulb and it's UL certified. There is an endless variety of 4" trims available from many manufacturers. For example, Juno Lighting offers 70 4" downlight trims and that's before counting the number of finishes available.

So in other words there are 4inch lights that will output the same amount of lumens as 6 inch ?

Don't worry about light output. Four-inch recessed lights put out plenty of light; you'll want to install dimmers to control the output. I installed Lightolier 3" recessed lights in a hallway and they are incredibly bright. Six-inch recessed lights are considered long out of style.

I agree , that’s why I’m trying to get 4 inch but the builder states that electrician can not find 4 inch light that will support 60 watt bulb.

You won't be using incandescent bulbs for your recessed lights but, nonetheless, these example housings will accept up to 75 watt bulbs. The important thing now is to have 4" housings installed and wired in appropriate locations in the ceiling.

• E26 Series 4 in. Aluminum New Construction IC Air-Tite Recessed Housing
• Juno IC1 New Construction Airtight IC Recessed Light Housing
• Juno TC1 New Construction Non-IC Recessed Light Housing

Remember that there are requirements for housings that will touch insulation and are installed in bathrooms. You would need different housings for sloped ceilings.

Trim (examples on page 2) is installed into a housing after it is installed in the ceiling. You can also install retrofit LED trim such as that linked earlier.

-- amateur