Definition df-iun 4012

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 4044. Theorem uniiun 4064 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 4010	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1401	. . . . 5 class y
7	6, 3	wcel 2209	. . . 4 wff y e. B
8	7, 1, 2	wrex 2529	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2224	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1402	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  4014  nfiunxy  4036  nfiunya  4038  nfiu1  4040  dfiunv2  4046  cbviun  4047  iunss  4051  uniiun  4064  iunopab  4422  opeliunxp  4828  reliun  4896  fnasrn  5881  fnasrng  5883  abrexex2g  6342  abrexex2  6346  bdciun  16821