Theorem qsss 8716

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 3445	. . . 4 ⊢ 𝑥 ∈ V
2	1	elqs 8705	. . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅)
3		qsss.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴)
4	3	ecss 8689	. . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴)
5		sseq1 3960	. . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴))
6	4, 5	syl5ibrcom 247	. . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴))
7		velpw 4560	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	6, 7	imbitrrdi 252	. . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
9	8	rexlimdvw 3143	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
10	2, 9	biimtrid 242	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
11	10	ssrdv 3940	1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3902  𝒫 cpw 4555   Er wer 8634  [cec 8635   / cqs 8636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-er 8637  df-ec 8639  df-qs 8643

This theorem is referenced by:  nrex1  10979  wuncn  11085  qshash  15754  lagsubg2  19127  lagsubg  19128  ghmqusnsg  19215  ghmquskerlem3  19219  ghmqusker  19220  orbsta2  19247  sylow1lem3  19533  sylow2alem2  19551  sylow2a  19552  sylow2blem2  19554  sylow2blem3  19555  sylow3lem3  19562  sylow3lem4  19563  rhmqusnsg  21244  vitalilem5  25573  vitali  25574  qerclwwlknfi  30152  lmhmqusker  33500  rhmquskerlem  33508  prjspnssbas  42931