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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 35700 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
2 | brcoss 35678 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴))) | |
3 | 2 | anidms 569 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴))) |
4 | pm4.24 566 | . . . 4 ⊢ (𝑢𝑅𝐴 ↔ (𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴)) | |
5 | 4 | exbii 1848 | . . 3 ⊢ (∃𝑢 𝑢𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴 ∧ 𝑢𝑅𝐴)) |
6 | 3, 5 | syl6bbr 291 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≀ 𝑅𝐴 ↔ ∃𝑢 𝑢𝑅𝐴)) |
7 | 1, 6 | bitr4d 284 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ≀ 𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1780 ∈ wcel 2114 class class class wbr 5068 dom cdm 5557 ≀ ccoss 35455 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-coss 35661 |
This theorem is referenced by: refrelcosslem 35704 |
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