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Mirrors > Home > ILE Home > Th. List > qsss | 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 2560 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elqs 6157 | . . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅) |
3 | qsss.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
4 | 3 | ecss 6147 | . . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴) |
5 | sseq1 2966 | . . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 146 | . . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴)) |
7 | selpw 3366 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
8 | 6, 7 | syl6ibr 151 | . . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
9 | 8 | rexlimdvw 2436 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
10 | 2, 9 | syl5bi 141 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴)) |
11 | 10 | ssrdv 2951 | 1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∈ wcel 1393 ∃wrex 2307 ⊆ wss 2917 𝒫 cpw 3359 Er wer 6103 [cec 6104 / cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-rel 4352 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: axcnex 6935 |
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