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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 8275). 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 35819 | . 2 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} | |
2 | elecALTV 35687 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)) | |
3 | 2 | el2v 3448 | . . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥) |
4 | elecALTV 35687 | . . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦)) | |
5 | 4 | el2v 3448 | . . . . 5 ⊢ (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦) |
6 | 3, 5 | anbi12i 629 | . . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
7 | 6 | exbii 1849 | . . 3 ⊢ (∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)) |
8 | 7 | opabbii 5097 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)} |
9 | 1, 8 | eqtr4i 2824 | 1 ⊢ ≀ 𝑅 = {〈𝑥, 𝑦〉 ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 {copab 5092 [cec 8270 ≀ ccoss 35613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ec 8274 df-coss 35819 |
This theorem is referenced by: coss0 35879 |
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