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Theorem ecelqsg 8768
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 2726 . . 3 [𝐵]𝑅 = [𝐵]𝑅
2 eceq1 8743 . . . 4 (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅)
32rspceeqv 3628 . . 3 ((𝐵𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
41, 3mpan2 688 . 2 (𝐵𝐴 → ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅)
5 ecexg 8709 . . . 4 (𝑅𝑉 → [𝐵]𝑅 ∈ V)
6 elqsg 8764 . . . 4 ([𝐵]𝑅 ∈ V → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
75, 6syl 17 . . 3 (𝑅𝑉 → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅))
87biimpar 477 . 2 ((𝑅𝑉 ∧ ∃𝑥𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
94, 8sylan2 592 1 ((𝑅𝑉𝐵𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  [cec 8703   / cqs 8704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711
This theorem is referenced by:  ecelqsi  8769  qliftlem  8794  erov  8810  eroprf  8811  sylow2a  19539  sylow2blem1  19540  sylow2blem2  19541  cldsubg  23970  tgjustr  28233
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