Theorem uniiun 3939

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 3811	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3888	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2201	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1353   {cab 2163   E.wrex 2456   U.cuni 3809   U_ciun 3886

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-rex 2461  df-uni 3810  df-iun 3888

This theorem is referenced by:  iunpwss  3977  truni  4114  iunpw  4479  reluni  4748  rnuni  5038  imauni  5758  hashuni  11482  tgidm  13436  unicld  13478  tgrest  13531  txbasval  13629