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 8697	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		abrexexg 7934	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V)
3	1, 2	eqeltrid 2838	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071  Vcvv 3475  [cec 8689   / cqs 8690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-rep 5281

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477  df-qs 8697

This theorem is referenced by:  qsex  8758  pstmval  32806  pstmxmet  32808  imaexALTV  37105