Theorem uniiun 3861

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 3733	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3810	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2161	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1331   {cab 2123   E.wrex 2415   U.cuni 3731   U_ciun 3808

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-rex 2420  df-uni 3732  df-iun 3810

This theorem is referenced by:  iunpwss  3899  truni  4035  iunpw  4396  reluni  4657  rnuni  4945  imauni  5655  hashuni  11244  tgidm  12232  unicld  12274  tgrest  12327  txbasval  12425