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Theorem reluni 5071
Description: The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)
Assertion
Ref Expression
reluni  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Distinct variable group:    x, A

Proof of Theorem reluni
StepHypRef Expression
1 uniiun 4332 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 5032 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 5069 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 249 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wral 2799   U.cuni 4200   U_ciun 4280   Rel wrel 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-v 3080  df-in 3444  df-ss 3451  df-uni 4201  df-iun 4282  df-rel 4956
This theorem is referenced by:  fununi  5593  tfrlem6  6952  wfrlem6  27874  frrlem5b  27918  frrlem6  27922  bnj1379  32157
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