Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relbigcup Structured version   Visualization version   Unicode version

Theorem relbigcup 30664
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 4942 . . 3  |-  Rel  ( _V  X.  _V )
2 reldif 4953 . . 3  |-  ( Rel  ( _V  X.  _V )  ->  Rel  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  o.  _E  )  (x)  _V ) ) ) )
31, 2ax-mp 5 . 2  |-  Rel  (
( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  o.  _E  )  (x)  _V )
) )
4 df-bigcup 30624 . . 3  |-  Bigcup  =  ( ( _V  X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  o.  _E  )  (x)  _V )
) )
54releqi 4918 . 2  |-  ( Rel  Bigcup  <->  Rel  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  o.  _E  )  (x)  _V ) ) ) )
63, 5mpbir 213 1  |-  Rel  Bigcup
Colors of variables: wff setvar class
Syntax hints:   _Vcvv 3045    \ cdif 3401    /_\ csymdif 3662    _E cep 4743    X. cxp 4832   ran crn 4835    o. ccom 4838   Rel wrel 4839    (x) ctxp 30596   Bigcupcbigcup 30600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-opab 4462  df-xp 4840  df-rel 4841  df-bigcup 30624
This theorem is referenced by:  brbigcup  30665  dfbigcup2  30666
  Copyright terms: Public domain W3C validator