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Theorem ecelqsi 6962
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 6961 . 2  |-  ( ( R  e.  _V  /\  B  e.  A )  ->  [ B ] R  e.  ( A /. R
) )
31, 2mpan 653 1  |-  ( B  e.  A  ->  [ B ] R  e.  ( A /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   _Vcvv 2958   [cec 6905   /.cqs 6906
This theorem is referenced by:  ecopqsi  6963  th3q  7015  0r  8957  1sr  8958  m1r  8959  addclsr  8960  mulclsr  8961  divseccl  14998  orbsta  15092  frgpeccl  15395  divstgphaus  18154  vitalilem2  19503  vitalilem3  19504  pstmfval  24293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-ec 6909  df-qs 6913
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