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Theorem dfcoss2 35541
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 8281). 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 35539 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 elecALTV 35408 . . . . . 6 ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥))
32el2v 3499 . . . . 5 (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥)
4 elecALTV 35408 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦))
54el2v 3499 . . . . 5 (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦)
63, 5anbi12i 626 . . . 4 ((𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
76exbii 1839 . . 3 (∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
87opabbii 5124 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
91, 8eqtr4i 2844 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492   class class class wbr 5057  {copab 5119  [cec 8276  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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  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-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280  df-coss 35539
This theorem is referenced by:  coss0  35599
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