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Theorem ecelqsi 6731
Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
ecelqsi.1  |-  R  e. 
_V
Assertion
Ref Expression
ecelqsi  |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )

Proof of Theorem ecelqsi
StepHypRef Expression
1 ecelqsi.1 . 2  |-  R  e. 
_V
2 ecelqsg 6730 . 2  |-  ( ( R  e.  _V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
31, 2mpan 651 1  |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696   _Vcvv 2801   [cec 6674   /.cqs 6675
This theorem is referenced by:  ecopqsi  6732  th3q  6783  0r  8718  1sr  8719  m1r  8720  addclsr  8721  mulclsr  8722  divseccl  14689  orbsta  14783  frgpeccl  15086  divstgphaus  17821  vitalilem2  18980  vitalilem3  18981
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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