Theorem uniiun 3981

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 3852	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3929	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2229	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1373   {cab 2191   E.wrex 2485   U.cuni 3850   U_ciun 3927

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-rex 2490  df-uni 3851  df-iun 3929

This theorem is referenced by:  iunpwss  4019  truni  4156  iunpw  4527  reluni  4798  rnuni  5094  imauni  5830  hashuni  11793  tgidm  14546  unicld  14588  tgrest  14641  txbasval  14739