Theorem uniiun 3919

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 3791	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3868	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2189	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1343   {cab 2151   E.wrex 2445   U.cuni 3789   U_ciun 3866

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-rex 2450  df-uni 3790  df-iun 3868

This theorem is referenced by:  iunpwss  3957  truni  4094  iunpw  4458  reluni  4727  rnuni  5015  imauni  5729  hashuni  11423  tgidm  12714  unicld  12756  tgrest  12809  txbasval  12907