Theorem uniiun 4019

Description: Class union in terms of indexed union. Definition in [Stoll] p. 43. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
uniiun	|- U.A = U_x e. A x

Distinct variable group:   x,A

Proof of Theorem uniiun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfuni2 3893	. 2 |- U.A = {y | E.x e. A y e. x}
2		df-iun 3971	. 2 |- U_x e. A x = {y | E.x e. A y e. x}
3	1, 2	eqtr4i 2376	1 |- U.A = U_x e. A x

Syntax hints:   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  U.cuni 3891  U_ciun 3969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-rex 2620  df-uni 3892  df-iun 3971

This theorem is referenced by:  iununi  4050  iunpwss  4055  imauni  5465