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Theorem dfcoss2 35731
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 8282). 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 35729 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 elecALTV 35597 . . . . . 6 ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥))
32el2v 3487 . . . . 5 (𝑥 ∈ [𝑢]𝑅𝑢𝑅𝑥)
4 elecALTV 35597 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦))
54el2v 3487 . . . . 5 (𝑦 ∈ [𝑢]𝑅𝑢𝑅𝑦)
63, 5anbi12i 629 . . . 4 ((𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
76exbii 1849 . . 3 (∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
87opabbii 5119 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
91, 8eqtr4i 2850 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅𝑦 ∈ [𝑢]𝑅)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480   class class class wbr 5052  {copab 5114  [cec 8277  ccoss 35523
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8281  df-coss 35729
This theorem is referenced by:  coss0  35789
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