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Mirrors > Home > MPE Home > Th. List > ecelqsg | Structured version Visualization version GIF version |
Description: Membership of an equivalence class in a quotient set. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
ecelqsg | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . 3 ⊢ [𝐵]𝑅 = [𝐵]𝑅 | |
2 | eceq1 8743 | . . . 4 ⊢ (𝑥 = 𝐵 → [𝑥]𝑅 = [𝐵]𝑅) | |
3 | 2 | rspceeqv 3628 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ [𝐵]𝑅 = [𝐵]𝑅) → ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) |
4 | 1, 3 | mpan2 688 | . 2 ⊢ (𝐵 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) |
5 | ecexg 8709 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → [𝐵]𝑅 ∈ V) | |
6 | elqsg 8764 | . . . 4 ⊢ ([𝐵]𝑅 ∈ V → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅)) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ([𝐵]𝑅 ∈ (𝐴 / 𝑅) ↔ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅)) |
8 | 7 | biimpar 477 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 [𝐵]𝑅 = [𝑥]𝑅) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
9 | 4, 8 | sylan2 592 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 [cec 8703 / cqs 8704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-xp 5675 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8707 df-qs 8711 |
This theorem is referenced by: ecelqsi 8769 qliftlem 8794 erov 8810 eroprf 8811 sylow2a 19539 sylow2blem1 19540 sylow2blem2 19541 cldsubg 23970 tgjustr 28233 |
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