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Theorem relelec 6904
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 2924 . . . 4  |-  ( A  e.  [ B ] R  ->  A  e.  _V )
2 ecexr 6869 . . . 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 4874 . . 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 6902 . 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 2916   class class class wbr 4172   Rel wrel 4842   [cec 6862
This theorem is referenced by:  eqgid  14947  tgptsmscls  18132  pstmfval  24244  topfneec  26261
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866
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