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Theorem qsss 6456
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 2663 . . . 4 𝑥 ∈ V
21elqs 6448 . . 3 (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦𝐴 𝑥 = [𝑦]𝑅)
3 qsss.1 . . . . . . 7 (𝜑𝑅 Er 𝐴)
43ecss 6438 . . . . . 6 (𝜑 → [𝑦]𝑅𝐴)
5 sseq1 3090 . . . . . 6 (𝑥 = [𝑦]𝑅 → (𝑥𝐴 ↔ [𝑦]𝑅𝐴))
64, 5syl5ibrcom 156 . . . . 5 (𝜑 → (𝑥 = [𝑦]𝑅𝑥𝐴))
7 velpw 3487 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
86, 7syl6ibr 161 . . . 4 (𝜑 → (𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
98rexlimdvw 2530 . . 3 (𝜑 → (∃𝑦𝐴 𝑥 = [𝑦]𝑅𝑥 ∈ 𝒫 𝐴))
102, 9syl5bi 151 . 2 (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
1110ssrdv 3073 1 (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  wrex 2394  wss 3041  𝒫 cpw 3480   Er wer 6394  [cec 6395   / cqs 6396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-er 6397  df-ec 6399  df-qs 6403
This theorem is referenced by:  axcnex  7635
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