Theorem qsexg 8757

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 8688	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		abrexexg 7946	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V)
3	1, 2	eqeltrid 2869	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457  [cec 8680   / cqs 8681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-rep 5232

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3090  df-v 3459  df-qs 8688

This theorem is referenced by:  qsex  8758  pstmval  34202  pstmxmet  34204  dmqsex  38873