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Theorem qsexg 8765
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 8705 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 abrexexg 7943 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} ∈ V)
31, 2eqeltrid 2837 1 (𝐴𝑉 → (𝐴 / 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {cab 2709  wrex 3070  Vcvv 3474  [cec 8697   / cqs 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-rep 5284
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-qs 8705
This theorem is referenced by:  qsex  8766  pstmval  32863  pstmxmet  32865  imaexALTV  37187
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