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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 3176 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elqs 7686 | . . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅) |
3 | qsss.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
4 | 3 | ecss 7675 | . . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴) |
5 | sseq1 3589 | . . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 236 | . . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴)) |
7 | selpw 4115 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
8 | 6, 7 | syl6ibr 241 | . . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
9 | 8 | rexlimdvw 3016 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
10 | 2, 9 | syl5bi 231 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴)) |
11 | 10 | ssrdv 3574 | 1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 𝒫 cpw 4108 Er wer 7626 [cec 7627 / cqs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-er 7629 df-ec 7631 df-qs 7635 |
This theorem is referenced by: axcnex 9847 wuncn 9870 qshash 14398 lagsubg2 17478 lagsubg 17479 orbsta2 17570 sylow1lem3 17838 sylow2alem2 17856 sylow2a 17857 sylow2blem2 17859 sylow2blem3 17860 sylow3lem3 17867 sylow3lem4 17868 vitalilem5 23187 vitali 23188 qerclwwlknfi 26357 qerclwwlksnfi 41257 |
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