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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoss2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8748). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.) | 
| Ref | Expression | 
|---|---|
| dfcoss2 | ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-coss 38412 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
| 2 | elecALTV 38267 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)) | |
| 3 | 2 | el2v 3487 | . . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥) | 
| 4 | elecALTV 38267 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦)) | |
| 5 | 4 | el2v 3487 | . . . . 5 ⊢ (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦) | 
| 6 | 3, 5 | anbi12i 628 | . . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) | 
| 7 | 6 | exbii 1848 | . . 3 ⊢ (∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) | 
| 8 | 7 | opabbii 5210 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | 
| 9 | 1, 8 | eqtr4i 2768 | 1 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 [cec 8743 ≀ ccoss 38182 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-coss 38412 | 
| This theorem is referenced by: coss0 38480 | 
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