Theorem uniiun 3926

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 3798	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3875	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2194	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1348   {cab 2156   E.wrex 2449   U.cuni 3796   U_ciun 3873

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-rex 2454  df-uni 3797  df-iun 3875

This theorem is referenced by:  iunpwss  3964  truni  4101  iunpw  4465  reluni  4734  rnuni  5022  imauni  5740  hashuni  11445  tgidm  12868  unicld  12910  tgrest  12963  txbasval  13061