Theorem uniiun 3970

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 3841	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3918	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2220	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1364   {cab 2182   E.wrex 2476   U.cuni 3839   U_ciun 3916

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-rex 2481  df-uni 3840  df-iun 3918

This theorem is referenced by:  iunpwss  4008  truni  4145  iunpw  4515  reluni  4786  rnuni  5081  imauni  5808  hashuni  11647  tgidm  14310  unicld  14352  tgrest  14405  txbasval  14503