Theorem qsexg 8770

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 8701	. 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 2864	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450  [cec 8693   / cqs 8694

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-rep 5231

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-qs 8701

This theorem is used by:  qsex  8771  pstmval  34461  pstmxmet  34463  dmqsex  39214