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Theorem relbigcup 24508
Description: The  Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.)
Assertion
Ref Expression
relbigcup  |-  Rel  Bigcup

Proof of Theorem relbigcup
StepHypRef Expression
1 relxp 4810 . . 3  |-  Rel  ( _V  X.  _V )
2 reldif 4821 . . 3  |-  ( Rel  ( _V  X.  _V )  ->  Rel  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V ) ) ) )
31, 2ax-mp 8 . 2  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V )
) )
4 df-bigcup 24470 . . 3  |-  Bigcup  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V )
) )
54releqi 4788 . 2  |-  ( Rel  Bigcup  <->  Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  o.  _E  )  (x)  _V )
) ) )
63, 5mpbir 200 1  |-  Rel  Bigcup
Colors of variables: wff set class
Syntax hints:   _Vcvv 2801    \ cdif 3162    _E cep 4319    X. cxp 4703   ran crn 4706    o. ccom 4709   Rel wrel 4710  (++)csymdif 24432    (x) ctxp 24444   Bigcupcbigcup 24448
This theorem is referenced by:  brbigcup  24509  dfbigcup2  24510
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-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-opab 4094  df-xp 4711  df-rel 4712  df-bigcup 24470
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