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Theorem qsss 8819
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 (𝜑𝑅 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.
StepHypRef Expression
1 vex 3483 . . . 4 𝑥 ∈ V
21elqs 8810 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 8794 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 4008 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 247 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 velpw 4604 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7imbitrrdi 252 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 3159 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9biimtrid 242 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3988 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  wrex 3069  wss 3950  𝒫 cpw 4599   Er wer 8743  [cec 8744   / cqs 8745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-er 8746  df-ec 8748  df-qs 8752
This theorem is referenced by:  nrex1  11105  wuncn  11211  qshash  15864  lagsubg2  19213  lagsubg  19214  ghmqusnsg  19301  ghmquskerlem3  19305  ghmqusker  19306  orbsta2  19333  sylow1lem3  19619  sylow2alem2  19637  sylow2a  19638  sylow2blem2  19640  sylow2blem3  19641  sylow3lem3  19648  sylow3lem4  19649  rhmqusnsg  21296  vitalilem5  25648  vitali  25649  qerclwwlknfi  30093  lmhmqusker  33446  rhmquskerlem  33454  prjspnssbas  42636
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