Theorem uniiun 4029

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 3900	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3977	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2255	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1398   {cab 2217   E.wrex 2512   U.cuni 3898   U_ciun 3975

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-rex 2517  df-uni 3899  df-iun 3977

This theorem is referenced by:  iunpwss  4067  truni  4206  iunpw  4583  reluni  4856  rnuni  5155  imauni  5912  hashuni  12106  tgidm  14868  unicld  14910  tgrest  14963  txbasval  15061