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Theorem ecelqsg 6896
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 2388 . . 3  |-  [ B ] R  =  [ B ] R
2 eceq1 6878 . . . . 5  |-  ( x  =  B  ->  [ x ] R  =  [ B ] R )
32eqeq2d 2399 . . . 4  |-  ( x  =  B  ->  ( [ B ] R  =  [ x ] R  <->  [ B ] R  =  [ B ] R
) )
43rspcev 2996 . . 3  |-  ( ( B  e.  A  /\  [ B ] R  =  [ B ] R
)  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
51, 4mpan2 653 . 2  |-  ( B  e.  A  ->  E. x  e.  A  [ B ] R  =  [
x ] R )
6 ecexg 6846 . . . 4  |-  ( R  e.  V  ->  [ B ] R  e.  _V )
7 elqsg 6893 . . . 4  |-  ( [ B ] R  e. 
_V  ->  ( [ B ] R  e.  ( A /. R )  <->  E. x  e.  A  [ B ] R  =  [
x ] R ) )
86, 7syl 16 . . 3  |-  ( R  e.  V  ->  ( [ B ] R  e.  ( A /. R
)  <->  E. x  e.  A  [ B ] R  =  [ x ] R
) )
98biimpar 472 . 2  |-  ( ( R  e.  V  /\  E. x  e.  A  [ B ] R  =  [
x ] R )  ->  [ B ] R  e.  ( A /. R ) )
105, 9sylan2 461 1  |-  ( ( R  e.  V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   _Vcvv 2900   [cec 6840   /.cqs 6841
This theorem is referenced by:  ecelqsi  6897  qliftlem  6922  erov  6938  eroprf  6939  sylow2a  15181  sylow2blem1  15182  sylow2blem2  15183  cldsubg  18062
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-cnv 4827  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-ec 6844  df-qs 6848
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