Theorem uniiun 3952

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 3823	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3900	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2211	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1363   {cab 2173   E.wrex 2466   U.cuni 3821   U_ciun 3898

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-rex 2471  df-uni 3822  df-iun 3900

This theorem is referenced by:  iunpwss  3990  truni  4127  iunpw  4492  reluni  4761  rnuni  5052  imauni  5775  hashuni  11504  tgidm  13870  unicld  13912  tgrest  13965  txbasval  14063