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Theorem qsss 6101
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 2554 . . . 4 x V
21elqs 6093 . . 3 (x (A / 𝑅) ↔ y A x = [y]𝑅)
3 qsss.1 . . . . . . 7 (φ𝑅 Er A)
43ecss 6083 . . . . . 6 (φ → [y]𝑅A)
5 sseq1 2960 . . . . . 6 (x = [y]𝑅 → (xA ↔ [y]𝑅A))
64, 5syl5ibrcom 146 . . . . 5 (φ → (x = [y]𝑅xA))
7 selpw 3358 . . . . 5 (x 𝒫 AxA)
86, 7syl6ibr 151 . . . 4 (φ → (x = [y]𝑅x 𝒫 A))
98rexlimdvw 2430 . . 3 (φ → (y A x = [y]𝑅x 𝒫 A))
102, 9syl5bi 141 . 2 (φ → (x (A / 𝑅) → x 𝒫 A))
1110ssrdv 2945 1 (φ → (A / 𝑅) ⊆ 𝒫 A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  wrex 2301  wss 2911  𝒫 cpw 3351   Er wer 6039  [cec 6040   / cqs 6041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044  df-qs 6048
This theorem is referenced by:  axcnex  6725
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