Definition df-iun 3929

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 3961. Theorem uniiun 3981 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 3927	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1372	. . . . 5 class y
7	6, 3	wcel 2176	. . . 4 wff y e. B
8	7, 1, 2	wrex 2485	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2191	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1373	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3931  nfiunxy  3953  nfiunya  3955  nfiu1  3957  dfiunv2  3963  cbviun  3964  iunss  3968  uniiun  3981  iunopab  4328  opeliunxp  4730  reliun  4796  fnasrn  5758  fnasrng  5760  abrexex2g  6205  abrexex2  6209  bdciun  15814