Theorem uniiun 4066

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 3937	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 4014	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2262	1 |- U. A = U_ x e. A x

This proof depends on syntax axioms:   = wceq 1402   {cab 2224   E.wrex 2529   U.cuni 3935   U_ciun 4012

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-rex 2534  df-uni 3936  df-iun 4014

This theorem is used by:  iunpwss  4104  truni  4243  iunpw  4626  reluni  4900  rnuni  5199  imauni  5967  hashuni  12249  tgidm  15175  unicld  15217  tgrest  15270  txbasval  15368