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Theorem brcoss 38432
Description: 𝐴 and 𝐵 are cosets by 𝑅: a binary relation. (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
brcoss ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem brcoss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5147 . . . 4 (𝑥 = 𝐴 → (𝑢𝑅𝑥𝑢𝑅𝐴))
2 breq2 5147 . . . 4 (𝑦 = 𝐵 → (𝑢𝑅𝑦𝑢𝑅𝐵))
31, 2bi2anan9 638 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
43exbidv 1921 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
5 df-coss 38412 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
64, 5brabga 5539 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108   class class class wbr 5143  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-rab 3437  df-v 3482  df-dif 3954  df-un 3956  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-coss 38412
This theorem is referenced by:  brcoss2  38433  brcoss3  38434  brcosscnvcoss  38435  cocossss  38437  br1cossres  38440  eldmcoss2  38460  brcosscnv  38473  trcoss  38483
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