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Theorem ecelqsi 6919
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 6918 . 2  |-  ( ( R  e.  _V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
31, 2mpan 652 1  |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1721   _Vcvv 2916   [cec 6862   /.cqs 6863
This theorem is referenced by:  ecopqsi  6920  th3q  6972  0r  8911  1sr  8912  m1r  8913  addclsr  8914  mulclsr  8915  divseccl  14951  orbsta  15045  frgpeccl  15348  divstgphaus  18105  vitalilem2  19454  vitalilem3  19455  pstmfval  24244
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ec 6866  df-qs 6870
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