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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldmcoss2 | Structured version Visualization version GIF version |
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.) |
Ref | Expression |
---|---|
eldmcoss2 | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmcoss 35578 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
2 | brcoss 35556 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴))) | |
3 | 2 | anidms 567 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴))) |
4 | pm4.24 564 | . . . 4 ⊢ (𝑢𝑅𝐴 ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴)) | |
5 | 4 | exbii 1839 | . . 3 ⊢ (∃𝑢 𝑢𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴)) |
6 | 3, 5 | syl6bbr 290 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢 𝑢𝑅𝐴)) |
7 | 1, 6 | bitr4d 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 ≀ ccoss 35334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-coss 35539 |
This theorem is referenced by: refrelcosslem 35582 |
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