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Theorem reluni 4808
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 3957 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4772 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4806 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 242 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wral 2545   U.cuni 3829   U_ciun 3907   Rel wrel 4694
This theorem is referenced by:  fununi  5282  tfrlem6  6394  wfrlem6  23663  frrlem5b  23688  frrlem6  23692  bnj1379  28131
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-v 2792  df-in 3161  df-ss 3168  df-uni 3830  df-iun 3909  df-rel 4696
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