Theorem elqsecl 8560

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 8557	. 2 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = [𝑥] ∼ ))
2		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
3		dfec2 8501	. . . . 5 ⊢ (𝑥 ∈ V → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
4	2, 3	mp1i 13	. . . 4 ⊢ (𝐵 ∈ 𝑋 → [𝑥] ∼ = {𝑦 ∣ 𝑥 ∼ 𝑦})
5	4	eqeq2d 2749	. . 3 ⊢ (𝐵 ∈ 𝑋 → (𝐵 = [𝑥] ∼ ↔ 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
6	5	rexbidv 3226	. 2 ⊢ (𝐵 ∈ 𝑋 → (∃𝑥 ∈ 𝑊 𝐵 = [𝑥] ∼ ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))
7	1, 6	bitrd 278	1 ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∣ 𝑥 ∼ 𝑦}))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  Vcvv 3432   class class class wbr 5074  [cec 8496   / cqs 8497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-qs 8504

This theorem is referenced by:  eclclwwlkn1  28439