Theorem elqsecl 8771

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	⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 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

Step	Hyp	Ref	Expression
1		elqsg 8768	. 2 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = [𝑥] ∼ ))
2		vex 3477	. . . . 5 ⊢ 𝑥 ∈ V
3		dfec2 8712	. . . . 5 ⊢ (𝑥 ∈ V → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
4	2, 3	mp1i 13	. . . 4 ⊢ (𝐵 ∈ 𝑋 → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
5	4	eqeq2d 2742	. . 3 ⊢ (𝐵 ∈ 𝑋 → (𝐵 = [𝑥] ∼ ↔ 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
6	5	rexbidv 3177	. 2 ⊢ (𝐵 ∈ 𝑋 → (∃𝑥 ∈ 𝑊 𝐵 = [𝑥] ∼ ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
7	1, 6	bitrd 279	1 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  {cab 2708  ∃wrex 3069  Vcvv 3473   class class class wbr 5148  [cec 8707   / cqs 8708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8711  df-qs 8715

This theorem is referenced by:  eclclwwlkn1  29761