Theorem qsexg 8767

Description: A quotient set exists. (Contributed by FL, 19-May-2007.) (Revised by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
qsexg	⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

Proof of Theorem qsexg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-qs 8698	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		abrexexg 7956	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V)
3	1, 2	eqeltrid 2866	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  {cab 2740  ∃wrex 3088  Vcvv 3454  [cec 8690   / cqs 8691

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-rep 5237

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-v 3456  df-qs 8698

This theorem is used by:  qsex  8768  pstmval  34294  pstmxmet  34296  dmqsex  39039