Theorem elqsg 6819

Description: Closed form of elqs 6820. (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 2239	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅))
2	1	rexbidv 2543	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
3		df-qs 6773	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	2, 3	elab2g 2964	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∃wrex 2521  [cec 6765   / cqs 6766

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-qs 6773

This theorem is referenced by:  elqs  6820  elqsi  6821  ecelqsg  6822  quselbasg  13947