Theorem uniiun 3966

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 3837	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3914	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2217	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1364   {cab 2179   E.wrex 2473   U.cuni 3835   U_ciun 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-rex 2478  df-uni 3836  df-iun 3914

This theorem is referenced by:  iunpwss  4004  truni  4141  iunpw  4511  reluni  4782  rnuni  5077  imauni  5804  hashuni  11625  tgidm  14242  unicld  14284  tgrest  14337  txbasval  14435