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Theorem elqsecl 8037
 Description: Membership in a quotient set by an equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)
Assertion
Ref Expression
elqsecl (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
Distinct variable groups:   𝑥, ,𝑦   𝑥,𝐵   𝑥,𝑊   𝑥,𝑋
Allowed substitution hints:   𝐵(𝑦)   𝑊(𝑦)   𝑋(𝑦)

Proof of Theorem elqsecl
StepHypRef Expression
1 elqsg 8034 . 2 (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = [𝑥] ))
2 vex 3386 . . . . 5 𝑥 ∈ V
3 dfec2 7983 . . . . 5 (𝑥 ∈ V → [𝑥] = {𝑦𝑥 𝑦})
42, 3mp1i 13 . . . 4 (𝐵𝑋 → [𝑥] = {𝑦𝑥 𝑦})
54eqeq2d 2807 . . 3 (𝐵𝑋 → (𝐵 = [𝑥] 𝐵 = {𝑦𝑥 𝑦}))
65rexbidv 3231 . 2 (𝐵𝑋 → (∃𝑥𝑊 𝐵 = [𝑥] ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
71, 6bitrd 271 1 (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  {cab 2783  ∃wrex 3088  Vcvv 3383   class class class wbr 4841  [cec 7978   / cqs 7979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-ec 7982  df-qs 7986 This theorem is referenced by:  eclclwwlkn1  27385
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