|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > qsss | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| qsss.1 | ⊢ (𝜑 → 𝑅 Er 𝐴) | 
| Ref | Expression | 
|---|---|
| qsss | ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 2 | 1 | elqs 8810 | . . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅) | 
| 3 | qsss.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
| 4 | 3 | ecss 8794 | . . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴) | 
| 5 | sseq1 4008 | . . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴)) | |
| 6 | 4, 5 | syl5ibrcom 247 | . . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴)) | 
| 7 | velpw 4604 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
| 8 | 6, 7 | imbitrrdi 252 | . . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) | 
| 9 | 8 | rexlimdvw 3159 | . . 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 3988 | 1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ⊆ wss 3950 𝒫 cpw 4599 Er wer 8743 [cec 8744 / cqs 8745 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-er 8746 df-ec 8748 df-qs 8752 | 
| This theorem is referenced by: nrex1 11105 wuncn 11211 qshash 15864 lagsubg2 19213 lagsubg 19214 ghmqusnsg 19301 ghmquskerlem3 19305 ghmqusker 19306 orbsta2 19333 sylow1lem3 19619 sylow2alem2 19637 sylow2a 19638 sylow2blem2 19640 sylow2blem3 19641 sylow3lem3 19648 sylow3lem4 19649 rhmqusnsg 21296 vitalilem5 25648 vitali 25649 qerclwwlknfi 30093 lmhmqusker 33446 rhmquskerlem 33454 prjspnssbas 42636 | 
| Copyright terms: Public domain | W3C validator |