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Theorem ecelqsg 7965
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecelqsg ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Proof of Theorem ecelqsg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2756 . . 3 [𝐵]𝑅 = [𝐵]𝑅
2 eceq1 7945 . . . . 5 (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅)
32eqeq2d 2766 . . . 4 (𝑥 = 𝐵 → ([𝐵]𝑅 = [𝑥]𝑅 ↔ [𝐵]𝑅 = [𝐵]𝑅))
43rspcev 3445 . . 3 ((𝐵𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
51, 4mpan2 709 . 2 (𝐵𝐴 → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
6 ecexg 7911 . . . 4 (𝑅𝑉 → [𝐵]𝑅 ∈ V)
7 elqsg 7961 . . . 4 ([𝐵]𝑅 ∈ V → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
86, 7syl 17 . . 3 (𝑅𝑉 → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
98biimpar 503 . 2 ((𝑅𝑉 ∧ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
105, 9sylan2 492 1 ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1628  wcel 2135  wrex 3047  Vcvv 3336  [cec 7905   / cqs 7906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-xp 5268  df-cnv 5270  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-ec 7909  df-qs 7913
This theorem is referenced by:  ecelqsi  7966  qliftlem  7991  erov  8007  eroprf  8008  sylow2a  18230  sylow2blem1  18231  sylow2blem2  18232  cldsubg  22111
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