MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reluni Unicode version

Theorem reluni 4796
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 3929 . . 3  |-  U. A  =  U_ x  e.  A  x
21releqi 4760 . 2  |-  ( Rel  U. A  <->  Rel  U_ x  e.  A  x )
3 reliun 4794 . 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 2518   U.cuni 3801   U_ciun 3879   Rel wrel 4666
This theorem is referenced by:  fununi  5254  tfrlem6  6366  wfrlem6  23630  frrlem5b  23655  frrlem6  23659  bnj1379  27912
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 2239
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 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ral 2523  df-rex 2524  df-v 2765  df-in 3134  df-ss 3141  df-uni 3802  df-iun 3881  df-rel 4676
  Copyright terms: Public domain W3C validator