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Theorem ecelqsg 6714
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecelqsg  |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )

Proof of Theorem ecelqsg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  [ B ] R  =  [ B ] R
2 eceq1 6696 . . . . 5  |-  ( x  =  B  ->  [ x ] R  =  [ B ] R )
32eqeq2d 2294 . . . 4  |-  ( x  =  B  ->  ( [ B ] R  =  [ x ] R  <->  [ B ] R  =  [ B ] R
) )
43rspcev 2884 . . 3  |-  ( ( B  e.  A  /\  [ B ] R  =  [ B ] R
)  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
51, 4mpan2 652 . 2  |-  ( B  e.  A  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
6 ecexg 6664 . . . 4  |-  ( R  e.  V  ->  [ B ] R  e.  _V )
7 elqsg 6711 . . . 4  |-  ( [ B ] R  e. 
_V  ->  ( [ B ] R  e.  ( A /. R )  <->  E. x  e.  A  [ B ] R  =  [
x ] R ) )
86, 7syl 15 . . 3  |-  ( R  e.  V  ->  ( [ B ] R  e.  ( A /. R
)  <->  E. x  e.  A  [ B ] R  =  [ x ] R
) )
98biimpar 471 . 2  |-  ( ( R  e.  V  /\  E. x  e.  A  [ B ] R  =  [
x ] R )  ->  [ B ] R  e.  ( A /. R ) )
105, 9sylan2 460 1  |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   [cec 6658   /.cqs 6659
This theorem is referenced by:  ecelqsi  6715  qliftlem  6739  erov  6755  eroprf  6756  sylow2a  14930  sylow2blem1  14931  sylow2blem2  14932  cldsubg  17793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-ec 6662  df-qs 6666
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