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Theorem brcoss 36304
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 5066 . . . 4 (𝑥 = 𝐴 → (𝑢𝑅𝑥𝑢𝑅𝐴))
2 breq2 5066 . . . 4 (𝑦 = 𝐵 → (𝑢𝑅𝑦𝑢𝑅𝐵))
31, 2bi2anan9 639 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
43exbidv 1929 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
5 df-coss 36287 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
64, 5brabga 5424 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wex 1787  wcel 2111   class class class wbr 5062  ccoss 36083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-br 5063  df-opab 5125  df-coss 36287
This theorem is referenced by:  brcoss2  36305  brcoss3  36306  brcosscnvcoss  36307  cocossss  36309  br1cossres  36312  eldmcoss2  36327  brcosscnv  36340  trcoss  36350
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