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Theorem reluni 5054
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 4313 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 5016 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 5052 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 249 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   A.wral 2746   U.cuni 4180   U_ciun 4260   Rel wrel 4935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-v 3053  df-in 3413  df-ss 3420  df-uni 4181  df-iun 4262  df-rel 4937
This theorem is referenced by:  fununi  5579  tfrlem6  6991  wfrlem6  29553  frrlem5b  29597  frrlem6  29601  bnj1379  34275
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