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

Theorem qsexg 8793
Description: A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
qsexg (𝐴𝑉 → (𝐴 / 𝑅) ∈ V)

Proof of Theorem qsexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-qs 8730 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 abrexexg 7964 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} ∈ V)
31, 2eqeltrid 2833 1 (𝐴𝑉 → (𝐴 / 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {cab 2705  wrex 3067  Vcvv 3471  [cec 8722   / cqs 8723
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 2699  ax-rep 5285
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3068  df-v 3473  df-qs 8730
This theorem is referenced by:  qsex  8794  pstmval  33496  pstmxmet  33498  imaexALTV  37802
  Copyright terms: Public domain W3C validator