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Theorem ecelqsi 8769
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 8768 . 2 ((𝑅 ∈ V ∧ 𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
31, 2mpan 686 1 (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  Vcvv 3472  [cec 8703   / cqs 8704
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8707  df-qs 8711
This theorem is referenced by:  ecopqsi  8770  addsrpr  11072  mulsrpr  11073  0r  11077  1sr  11078  m1r  11079  addclsr  11080  mulclsr  11081  quseccl0  19100  orbsta  19218  frgpeccl  19670  rngqiprngimf  21056  qustgphaus  23847  vitalilem2  25358  vitalilem3  25359  nsgqusf1olem1  32798  ghmquskerlem1  32802  ghmquskerco  32803  ghmqusker  32806  qsidomlem1  32845  qsdrngilem  32882  qsdrngi  32883  qsdrnglem2  32884  pstmfval  33174
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