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Theorem ecelqsi 7955
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 7954 . 2 ((𝑅 ∈ V ∧ 𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
31, 2mpan 670 1 (𝐵𝐴 → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  Vcvv 3351  [cec 7894   / cqs 7895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ec 7898  df-qs 7902
This theorem is referenced by:  ecopqsi  7956  addsrpr  10098  mulsrpr  10099  0r  10103  1sr  10104  m1r  10105  addclsr  10106  mulclsr  10107  quseccl  17858  orbsta  17953  frgpeccl  18381  qustgphaus  22146  vitalilem2  23597  vitalilem3  23598  pstmfval  30279
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