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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 3478 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elqs 8759 | . . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅) |
3 | qsss.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
4 | 3 | ecss 8745 | . . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴) |
5 | sseq1 4006 | . . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 246 | . . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴)) |
7 | velpw 4606 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
8 | 6, 7 | syl6ibr 251 | . . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
9 | 8 | rexlimdvw 3160 | . . 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 3987 | 1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3947 𝒫 cpw 4601 Er wer 8696 [cec 8697 / cqs 8698 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-er 8699 df-ec 8701 df-qs 8705 |
This theorem is referenced by: nrex1 11055 wuncn 11161 qshash 15769 lagsubg2 19065 lagsubg 19066 orbsta2 19172 sylow1lem3 19462 sylow2alem2 19480 sylow2a 19481 sylow2blem2 19483 sylow2blem3 19484 sylow3lem3 19491 sylow3lem4 19492 vitalilem5 25120 vitali 25121 qerclwwlknfi 29315 ghmquskerlem3 32519 ghmqusker 32520 lmhmqusker 32522 rhmquskerlem 32531 prjspnssbas 41359 |
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