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Theorem qsss 8460
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 3412 . . . 4 𝑥 ∈ V
21elqs 8451 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 8437 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 3926 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 250 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 velpw 4518 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7syl6ibr 255 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 3209 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9syl5bi 245 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3907 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wrex 3062  wss 3866  𝒫 cpw 4513   Er wer 8388  [cec 8389   / cqs 8390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-er 8391  df-ec 8393  df-qs 8397
This theorem is referenced by:  nrex1  10678  wuncn  10784  qshash  15391  lagsubg2  18605  lagsubg  18606  orbsta2  18708  sylow1lem3  18989  sylow2alem2  19007  sylow2a  19008  sylow2blem2  19010  sylow2blem3  19011  sylow3lem3  19018  sylow3lem4  19019  vitalilem5  24509  vitali  24510  qerclwwlknfi  28156
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