Theorem uniiun 4024

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 3895	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3972	. 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 1397   {cab 2217   E.wrex 2511   U.cuni 3893   U_ciun 3970

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-rex 2516  df-uni 3894  df-iun 3972

This theorem is referenced by:  iunpwss  4062  truni  4201  iunpw  4577  reluni  4850  rnuni  5148  imauni  5901  hashuni  12042  tgidm  14797  unicld  14839  tgrest  14892  txbasval  14990