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Theorem brcoss 37301
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 5153 . . . 4 (𝑥 = 𝐴 → (𝑢𝑅𝑥𝑢𝑅𝐴))
2 breq2 5153 . . . 4 (𝑦 = 𝐵 → (𝑢𝑅𝑦𝑢𝑅𝐵))
31, 2bi2anan9 638 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
43exbidv 1925 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
5 df-coss 37281 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
64, 5brabga 5535 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107   class class class wbr 5149  ccoss 37043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-coss 37281
This theorem is referenced by:  brcoss2  37302  brcoss3  37303  brcosscnvcoss  37304  cocossss  37306  br1cossres  37309  eldmcoss2  37329  brcosscnv  37342  trcoss  37352
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