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Theorem reluni 4983
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 4131 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4946 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4981 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 241 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   A.wral 2692   U.cuni 4002   U_ciun 4080   Rel wrel 4869
This theorem is referenced by:  fununi  5503  tfrlem6  6629  wfrlem6  25511  frrlem5b  25536  frrlem6  25540  bnj1379  28954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ral 2697  df-rex 2698  df-v 2945  df-in 3314  df-ss 3321  df-uni 4003  df-iun 4082  df-rel 4871
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