Theorem uniiun 4018

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 3889	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3966	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2253	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1395   {cab 2215   E.wrex 2509   U.cuni 3887   U_ciun 3964

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-rex 2514  df-uni 3888  df-iun 3966

This theorem is referenced by:  iunpwss  4056  truni  4195  iunpw  4570  reluni  4841  rnuni  5139  imauni  5884  hashuni  11988  tgidm  14742  unicld  14784  tgrest  14837  txbasval  14935