Theorem qsss 8776

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	⊢ (𝜑 → 𝑅 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.

Step	Hyp	Ref	Expression
1		vex 3476	. . . 4 ⊢ 𝑥 ∈ V
2	1	elqs 8767	. . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅)
3		qsss.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴)
4	3	ecss 8753	. . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴)
5		sseq1 4008	. . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴))
6	4, 5	syl5ibrcom 246	. . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴))
7		velpw 4608	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	6, 7	imbitrrdi 251	. . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
9	8	rexlimdvw 3158	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
10	2, 9	biimtrid 241	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
11	10	ssrdv 3989	1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  ∃wrex 3068   ⊆ wss 3949  𝒫 cpw 4603   Er wer 8704  [cec 8705   / cqs 8706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-er 8707  df-ec 8709  df-qs 8713

This theorem is referenced by:  nrex1  11063  wuncn  11169  qshash  15779  lagsubg2  19111  lagsubg  19112  orbsta2  19221  sylow1lem3  19511  sylow2alem2  19529  sylow2a  19530  sylow2blem2  19532  sylow2blem3  19533  sylow3lem3  19540  sylow3lem4  19541  vitalilem5  25363  vitali  25364  qerclwwlknfi  29591  ghmquskerlem3  32803  ghmqusker  32804  lmhmqusker  32806  rhmquskerlem  32815  prjspnssbas  41667