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Theorem qsss 8336
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 3476 . . . 4 𝑥 ∈ V
21elqs 8327 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 8313 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 3971 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 249 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 velpw 4520 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7syl6ibr 254 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 3277 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9syl5bi 244 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3952 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  wrex 3126  wss 3913  𝒫 cpw 4515   Er wer 8264  [cec 8265   / cqs 8266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-br 5043  df-opab 5105  df-xp 5537  df-rel 5538  df-cnv 5539  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-er 8267  df-ec 8269  df-qs 8273
This theorem is referenced by:  nrex1  10464  wuncn  10570  qshash  15162  lagsubg2  18320  lagsubg  18321  orbsta2  18423  sylow1lem3  18704  sylow2alem2  18722  sylow2a  18723  sylow2blem2  18725  sylow2blem3  18726  sylow3lem3  18733  sylow3lem4  18734  vitalilem5  24195  vitali  24196  qerclwwlknfi  27837
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