Theorem uniiun 3995

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 3866	. 2 |- U. A = { y | E. x e. A y e. x }
2		df-iun 3943	. 2 |- U_ x e. A x = { y | E. x e. A y e. x }
3	1, 2	eqtr4i 2231	1 |- U. A = U_ x e. A x

Syntax hints:   = wceq 1373   {cab 2193   E.wrex 2487   U.cuni 3864   U_ciun 3941

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-rex 2492  df-uni 3865  df-iun 3943

This theorem is referenced by:  iunpwss  4033  truni  4172  iunpw  4545  reluni  4816  rnuni  5113  imauni  5853  hashuni  11908  tgidm  14661  unicld  14703  tgrest  14756  txbasval  14854