Theorem elqsecl 8717

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 8714	. 2 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = [𝑥] ∼ ))
2		vex 3448	. . . . 5 ⊢ 𝑥 ∈ V
3		dfec2 8651	. . . . 5 ⊢ (𝑥 ∈ V → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
4	2, 3	mp1i 13	. . . 4 ⊢ (𝐵 ∈ 𝑋 → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
5	4	eqeq2d 2740	. . 3 ⊢ (𝐵 ∈ 𝑋 → (𝐵 = [𝑥] ∼ ↔ 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
6	5	rexbidv 3157	. 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 206   = wceq 1540   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3444   class class class wbr 5102  [cec 8646   / cqs 8647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654

This theorem is referenced by:  eclclwwlkn1  30054