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Theorem reluni 4975
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 4352 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4937 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4973 . 2  |-  ( Rel  U_ x  e.  A  x 
<-> 
A. x  e.  A  Rel  x )
42, 3bitri 252 1  |-  ( Rel  U. A  <->  A. x  e.  A  Rel  x )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187   A.wral 2771   U.cuni 4219   U_ciun 4299   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-in 3443  df-ss 3450  df-uni 4220  df-iun 4301  df-rel 4860
This theorem is referenced by:  fununi  5667  wfrrel  7052  tfrlem6  7111  bnj1379  29650  frrlem5b  30526  frrlem6  30530
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