Theorem elqsg 6602

Description: Closed form of elqs 6603. (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 2195	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅 ↔ 𝐵 = [𝑥]𝑅))
2	1	rexbidv 2490	. 2 ⊢ (𝑦 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))
3		df-qs 6558	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
4	2, 3	elab2g 2898	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥]𝑅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1363   ∈ wcel 2159  ∃wrex 2468  [cec 6550   / cqs 6551

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-rex 2473  df-v 2753  df-qs 6558

This theorem is referenced by:  elqs  6603  elqsi  6604  ecelqsg  6605  quselbasg  13134