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Theorem qsexg 8815
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 8751 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
2 abrexexg 7985 . 2 (𝐴𝑉 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} ∈ V)
31, 2eqeltrid 2845 1 (𝐴𝑉 → (𝐴 / 𝑅) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-rep 5279
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-qs 8751
This theorem is referenced by:  qsex  8816  pstmval  33894  pstmxmet  33896  imaexALTV  38331
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