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Theorem qsss 8768
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 3478 . . . 4 𝑥 ∈ V
21elqs 8759 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 8745 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 4006 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 246 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 velpw 4606 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7syl6ibr 251 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 3160 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9biimtrid 241 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3987 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wrex 3070  wss 3947  𝒫 cpw 4601   Er wer 8696  [cec 8697   / cqs 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-er 8699  df-ec 8701  df-qs 8705
This theorem is referenced by:  nrex1  11055  wuncn  11161  qshash  15769  lagsubg2  19065  lagsubg  19066  orbsta2  19172  sylow1lem3  19462  sylow2alem2  19480  sylow2a  19481  sylow2blem2  19483  sylow2blem3  19484  sylow3lem3  19491  sylow3lem4  19492  vitalilem5  25120  vitali  25121  qerclwwlknfi  29315  ghmquskerlem3  32519  ghmqusker  32520  lmhmqusker  32522  rhmquskerlem  32531  prjspnssbas  41359
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