MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elqsg Structured version   Visualization version   GIF version

Theorem elqsg 8789
Description: Closed form of elqs 8790. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
elqsg (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 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.
StepHypRef Expression
1 eqeq1 2730 . . 3 (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅𝐵 = [𝑥]𝑅))
21rexbidv 3169 . 2 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
3 df-qs 8732 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
42, 3elab2g 3667 1 (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wrex 3060  [cec 8724   / cqs 8725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rex 3061  df-qs 8732
This theorem is referenced by:  elqs  8790  elqsi  8791  elqsecl  8792  ecelqsg  8793  quselbas  19174  ghmqusnsglem2  19271  ghmquskerlem2  19275  rngqiprngfulem1  21296  elpi1  25060  eldmqsres  37998  eldmqs1cossres  38370  prtlem11  38577
  Copyright terms: Public domain W3C validator