Definition df-iun 3815

Description: Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, A is independent of x (although this is not required by the definition), and B depends on x i.e. can be read informally as B ( x ). We call x the index, A the index set, and B the indexed set. In most books, x e. A is written as a subscript or underneath a union symbol U.. We use a special union symbol U_ to make it easier to distinguish from plain class union. In many theorems, you will see that x and A are in the same disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint variable group (meaning that can be thought of as B ( x ) i.e. can be substituted with a class expression containing x). An alternate definition tying indexed union to ordinary union is dfiun2 3847. Theorem uniiun 3866 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
df-iun	|- U_ x e. A B = { y | E. x e. A y e. B }

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 3813	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1330	. . . . 5 class y
7	6, 3	wcel 1480	. . . 4 wff y e. B
8	7, 1, 2	wrex 2417	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2125	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1331	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3817  nfiunxy  3839  nfiunya  3841  nfiu1  3843  dfiunv2  3849  cbviun  3850  iunss  3854  uniiun  3866  iunopab  4203  opeliunxp  4594  reliun  4660  fnasrn  5598  fnasrng  5600  abrexex2g  6018  abrexex2  6022  bdciun  13076