Theorem uniiun 3874

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 3746	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3823	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2164	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1332   {cab 2126   E.wrex 2418   U.cuni 3744   U_ciun 3821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-rex 2423  df-uni 3745  df-iun 3823

This theorem is referenced by:  iunpwss  3912  truni  4048  iunpw  4409  reluni  4670  rnuni  4958  imauni  5670  hashuni  11283  tgidm  12282  unicld  12324  tgrest  12377  txbasval  12475