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Theorem reluni 4978
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 4345 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4940 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4976 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 257 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   A.wral 2749   U.cuni 4212   U_ciun 4292   Rel wrel 4861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ral 2754  df-rex 2755  df-v 3059  df-in 3423  df-ss 3430  df-uni 4213  df-iun 4294  df-rel 4863
This theorem is referenced by:  fununi  5675  wfrrel  7072  tfrlem6  7131  bnj1379  29692  frrlem5b  30569  frrlem6  30573
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