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Theorem uniiun 3834
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.
StepHypRef Expression
1 dfuni2 3706 . 2  |-  U. A  =  { y  |  E. x  e.  A  y  e.  x }
2 df-iun 3783 . 2  |-  U_ x  e.  A  x  =  { y  |  E. x  e.  A  y  e.  x }
31, 2eqtr4i 2139 1  |-  U. A  =  U_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1314   {cab 2101   E.wrex 2392   U.cuni 3704   U_ciun 3781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-rex 2397  df-uni 3705  df-iun 3783
This theorem is referenced by:  iunpwss  3872  truni  4008  iunpw  4369  reluni  4630  rnuni  4918  imauni  5628  hashuni  11191  tgidm  12138  unicld  12180  tgrest  12233  txbasval  12331
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