Theorem qsss 8796

Description: A quotient set is a set of subsets of the base set. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)

Hypothesis

Ref	Expression
qsss.1	⊢ (𝜑 → 𝑅 Er 𝐴)

Assertion

Ref	Expression
qsss	⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Proof of Theorem qsss

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

Step	Hyp	Ref	Expression
1		vex 3454	. . . 4 ⊢ 𝑥 ∈ V
2	1	elqs 8785	. . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅)
3		qsss.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴)
4	3	ecss 8769	. . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴)
5		sseq1 3933	. . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴))
6	4, 5	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴))
7		velpw 4561	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	6, 7	imbitrrdi 255	. . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
9	8	rexlimdvw 3168	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
10	2, 9	biimtrid 245	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
11	10	ssrdv 3914	1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  𝒫 cpw 4556   Er wer 8714  [cec 8715   / cqs 8716

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-er 8717  df-ec 8719  df-qs 8723

This theorem is used by:  nrex1  11149  wuncn  11255  qshash  15994  lagsubg2  19409  lagsubg  19410  ghmqusnsg  19496  ghmquskerlem3  19500  ghmqusker  19501  orbsta2  19528  sylow1lem3  19814  sylow2alem2  19832  sylow2a  19833  sylow2blem2  19835  sylow2blem3  19836  sylow3lem3  19843  sylow3lem4  19844  rhmqusnsg  21581  vitalilem5  25933  vitali  25934  qerclwwlknfi  30664  lmhmqusker  33968  rhmquskerlem  33975  prjspnssbas  43650