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Theorem ecelqsi 8813
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 ∈ V
Assertion
Ref Expression
ecelqsi (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Proof of Theorem ecelqsi
StepHypRef Expression
1 ecelqsi.1 . 2 𝑅 ∈ V
2 ecelqsg 8812 . 2 ((𝑅 ∈ V ∧ 𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
31, 2mpan 690 1 (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  ecopqsi  8814  addsrpr  11115  mulsrpr  11116  0r  11120  1sr  11121  m1r  11122  addclsr  11123  mulclsr  11124  quseccl0  19203  ghmqusnsglem1  19298  ghmquskerlem1  19301  ghmquskerco  19302  ghmqusker  19305  orbsta  19331  frgpeccl  19779  rngqiprngimf  21307  qustgphaus  24131  vitalilem2  25644  vitalilem3  25645  rloccring  33274  rloc0g  33275  rloc1r  33276  rlocf1  33277  fracfld  33310  nsgqusf1olem1  33441  qsidomlem1  33480  qsdrngilem  33522  qsdrngi  33523  qsdrnglem2  33524  zringfrac  33582  pstmfval  33895
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