Theorem elqsg 6753

Description: Closed form of elqs 6754. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Assertion

Ref	Expression
elqsg	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elqsg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2238	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅))
2	1	rexbidv 2533	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
3		df-qs 6707	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	2, 3	elab2g 2953	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  [cec 6699   / cqs 6700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-qs 6707

This theorem is referenced by:  elqs  6754  elqsi  6755  ecelqsg  6756  quselbasg  13816