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

Theorem relelec 6912
Description: Membership in an equivalence class when  R is a relation. (Contributed by Mario Carneiro, 11-Sep-2015.)
Assertion
Ref Expression
relelec  |-  ( Rel 
R  ->  ( A  e.  [ B ] R  <->  B R A ) )

Proof of Theorem relelec
StepHypRef Expression
1 elex 2932 . . . 4  |-  ( A  e.  [ B ] R  ->  A  e.  _V )
2 ecexr 6877 . . . 4  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
31, 2jca 519 . . 3  |-  ( A  e.  [ B ] R  ->  ( A  e. 
_V  /\  B  e.  _V ) )
43adantl 453 . 2  |-  ( ( Rel  R  /\  A  e.  [ B ] R
)  ->  ( A  e.  _V  /\  B  e. 
_V ) )
5 brrelex12 4882 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 439 . 2  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
7 elecg 6910 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
84, 6, 7pm5.21nd 869 1  |-  ( Rel 
R  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   _Vcvv 2924   class class class wbr 4180   Rel wrel 4850   [cec 6870
This theorem is referenced by:  eqgid  14955  tgptsmscls  18140  pstmfval  24252  topfneec  26269
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ec 6874
  Copyright terms: Public domain W3C validator