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Theorem ecelqsg 6730
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 2296 . . 3  |-  [ B ] R  =  [ B ] R
2 eceq1 6712 . . . . 5  |-  ( x  =  B  ->  [ x ] R  =  [ B ] R )
32eqeq2d 2307 . . . 4  |-  ( x  =  B  ->  ( [ B ] R  =  [ x ] R  <->  [ B ] R  =  [ B ] R
) )
43rspcev 2897 . . 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 6680 . . . 4  |-  ( R  e.  V  ->  [ B ] R  e.  _V )
7 elqsg 6727 . . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   [cec 6674   /.cqs 6675
This theorem is referenced by:  ecelqsi  6731  qliftlem  6755  erov  6771  eroprf  6772  sylow2a  14946  sylow2blem1  14947  sylow2blem2  14948  cldsubg  17809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-ec 6678  df-qs 6682
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