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Theorem qsss 7975
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 3343 . . . 4 𝑥 ∈ V
21elqs 7966 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 7955 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 3767 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 237 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 selpw 4309 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7syl6ibr 242 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 3172 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9syl5bi 232 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3750 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wrex 3051  wss 3715  𝒫 cpw 4302   Er wer 7908  [cec 7909   / cqs 7910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-er 7911  df-ec 7913  df-qs 7917
This theorem is referenced by:  axcnex  10160  wuncn  10183  qshash  14758  lagsubg2  17856  lagsubg  17857  orbsta2  17947  sylow1lem3  18215  sylow2alem2  18233  sylow2a  18234  sylow2blem2  18236  sylow2blem3  18237  sylow3lem3  18244  sylow3lem4  18245  vitalilem5  23580  vitali  23581  qerclwwlknfi  27204
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