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Theorem uniiun 3710
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 3582 . 2  |-  U. A  =  { y  |  E. x  e.  A  y  e.  x }
2 df-iun 3659 . 2  |-  U_ x  e.  A  x  =  { y  |  E. x  e.  A  y  e.  x }
31, 2eqtr4i 2063 1  |-  U. A  =  U_ x  e.  A  x
Colors of variables: wff set class
Syntax hints:    = wceq 1243   {cab 2026   E.wrex 2307   U.cuni 3580   U_ciun 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-rex 2312  df-uni 3581  df-iun 3659
This theorem is referenced by:  iunpwss  3743  truni  3868  iunpw  4211  reluni  4460  rnuni  4735  imauni  5400
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