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Theorem relelec 6948
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 2966 . . . 4  |-  ( A  e.  [ B ] R  ->  A  e.  _V )
2 ecexr 6913 . . . 4  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
31, 2jca 520 . . 3  |-  ( A  e.  [ B ] R  ->  ( A  e. 
_V  /\  B  e.  _V ) )
43adantl 454 . 2  |-  ( ( Rel  R  /\  A  e.  [ B ] R
)  ->  ( A  e.  _V  /\  B  e. 
_V ) )
5 brrelex12 4918 . . 3  |-  ( ( Rel  R  /\  B R A )  ->  ( B  e.  _V  /\  A  e.  _V ) )
65ancomd 440 . 2  |-  ( ( Rel  R  /\  B R A )  ->  ( A  e.  _V  /\  B  e.  _V ) )
7 elecg 6946 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  e.  [ B ] R  <->  B R A ) )
84, 6, 7pm5.21nd 870 1  |-  ( Rel 
R  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   _Vcvv 2958   class class class wbr 4215   Rel wrel 4886   [cec 6906
This theorem is referenced by:  eqgid  14997  tgptsmscls  18184  pstmfval  24296  topfneec  26385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-ec 6910
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