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Theorem elqsecl 7957
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 7954 . 2 (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = [𝑥] ))
2 vex 3354 . . . . 5 𝑥 ∈ V
3 dfec2 7903 . . . . 5 (𝑥 ∈ V → [𝑥] = {𝑦𝑥 𝑦})
42, 3mp1i 13 . . . 4 (𝐵𝑋 → [𝑥] = {𝑦𝑥 𝑦})
54eqeq2d 2781 . . 3 (𝐵𝑋 → (𝐵 = [𝑥] 𝐵 = {𝑦𝑥 𝑦}))
65rexbidv 3200 . 2 (𝐵𝑋 → (∃𝑥𝑊 𝐵 = [𝑥] ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
71, 6bitrd 268 1 (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑥 𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351   class class class wbr 4787  [cec 7898   / cqs 7899
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
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-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ec 7902  df-qs 7906
This theorem is referenced by:  eclclwwlkn1  27233
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