Theorem qsexg 8814

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 8750	. 2 ⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅}
2		abrexexg 7984	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} ∈ V)
3	1, 2	eqeltrid 2843	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 / 𝑅) ∈ V)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068  Vcvv 3478  [cec 8742   / cqs 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-rep 5285

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480  df-qs 8750

This theorem is referenced by:  qsex  8815  pstmval  33856  pstmxmet  33858  imaexALTV  38312