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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 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elqs 8349 | . . 3 ⊢ (𝑥 ∈ (𝐴 / 𝑅) ↔ ∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅) |
3 | qsss.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Er 𝐴) | |
4 | 3 | ecss 8335 | . . . . . 6 ⊢ (𝜑 → [𝑦]𝑅 ⊆ 𝐴) |
5 | sseq1 3992 | . . . . . 6 ⊢ (𝑥 = [𝑦]𝑅 → (𝑥 ⊆ 𝐴 ↔ [𝑦]𝑅 ⊆ 𝐴)) | |
6 | 4, 5 | syl5ibrcom 249 | . . . . 5 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ⊆ 𝐴)) |
7 | velpw 4544 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
8 | 6, 7 | syl6ibr 254 | . . . 4 ⊢ (𝜑 → (𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
9 | 8 | rexlimdvw 3290 | . . 3 ⊢ (𝜑 → (∃𝑦 ∈ 𝐴 𝑥 = [𝑦]𝑅 → 𝑥 ∈ 𝒫 𝐴)) |
10 | 2, 9 | syl5bi 244 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 / 𝑅) → 𝑥 ∈ 𝒫 𝐴)) |
11 | 10 | ssrdv 3973 | 1 ⊢ (𝜑 → (𝐴 / 𝑅) ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ⊆ wss 3936 𝒫 cpw 4539 Er wer 8286 [cec 8287 / cqs 8288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-er 8289 df-ec 8291 df-qs 8295 |
This theorem is referenced by: nrex1 10486 wuncn 10592 qshash 15182 lagsubg2 18341 lagsubg 18342 orbsta2 18444 sylow1lem3 18725 sylow2alem2 18743 sylow2a 18744 sylow2blem2 18746 sylow2blem3 18747 sylow3lem3 18754 sylow3lem4 18755 vitalilem5 24213 vitali 24214 qerclwwlknfi 27852 |
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