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Theorem dfcoss2 38414
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.)
Assertion
Ref Expression
dfcoss2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem dfcoss2
StepHypRef Expression
1 df-coss 38412 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 elecALTV 38267 . . . . . 6 ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥))
32el2v 3487 . . . . 5 (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥)
4 elecALTV 38267 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦))
54el2v 3487 . . . . 5 (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦)
63, 5anbi12i 628 . . . 4 ((𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
76exbii 1848 . . 3 (∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
87opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
91, 8eqtr4i 2768 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [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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