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Theorem reluni 4761
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 3896 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4725 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4759 . 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 2516   U.cuni 3768   U_ciun 3846   Rel wrel 4631
This theorem is referenced by:  fununi  5219  tfrlem6  6331  wfrlem6  23595  frrlem5b  23620  frrlem6  23624  bnj1379  27875
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-rex 2521  df-v 2742  df-in 3101  df-ss 3108  df-uni 3769  df-iun 3848  df-rel 4641
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