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Theorem brcoss 38413
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 5152 . . . 4 (𝑥 = 𝐴 → (𝑢𝑅𝑥𝑢𝑅𝐴))
2 breq2 5152 . . . 4 (𝑦 = 𝐵 → (𝑢𝑅𝑦𝑢𝑅𝐵))
31, 2bi2anan9 638 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
43exbidv 1919 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
5 df-coss 38393 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
64, 5brabga 5544 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106   class class class wbr 5148  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-coss 38393
This theorem is referenced by:  brcoss2  38414  brcoss3  38415  brcosscnvcoss  38416  cocossss  38418  br1cossres  38421  eldmcoss2  38441  brcosscnv  38454  trcoss  38464
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