Theorem qsss 6351

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 2622	. . . 4 ⊢ 𝑥 ∈ V
2	1	elqs 6343	. . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅)
3		qsss.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴)
4	3	ecss 6333	. . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴)
5		sseq1 3047	. . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴))
6	4, 5	syl5ibrcom 155	. . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴))
7		selpw 3436	. . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	6, 7	syl6ibr 160	. . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
9	8	rexlimdvw 2492	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴))
10	2, 9	syl5bi 150	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴))
11	10	ssrdv 3031	1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   = wceq 1289   ∈ wcel 1438  ∃wrex 2360   ⊆ wss 2999  𝒫 cpw 3429   Er wer 6289  [cec 6290   / cqs 6291

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6292  df-ec 6294  df-qs 6298

This theorem is referenced by:  axcnex  7396