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Theorem ecelqsi 7751
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 7750 . 2 ((𝑅 ∈ V ∧ 𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
31, 2mpan 705 1 (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Vcvv 3186  [cec 7688   / cqs 7689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-xp 5082  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-ec 7692  df-qs 7696
This theorem is referenced by:  ecopqsi  7752  addsrpr  9843  mulsrpr  9844  0r  9848  1sr  9849  m1r  9850  addclsr  9851  mulclsr  9852  quseccl  17574  orbsta  17670  frgpeccl  18098  qustgphaus  21839  vitalilem2  23291  vitalilem3  23292  pstmfval  29733
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