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Theorem reluni 4824
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 3971 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4788 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4822 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 240 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   A.wral 2556   U.cuni 3843   U_ciun 3921   Rel wrel 4710
This theorem is referenced by:  fununi  5332  tfrlem6  6414  wfrlem6  24332  frrlem5b  24357  frrlem6  24361  bnj1379  29179
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844  df-iun 3923  df-rel 4712
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