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Theorem brcoss 38142
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 5149 . . . 4 (𝑥 = 𝐴 → (𝑢𝑅𝑥𝑢𝑅𝐴))
2 breq2 5149 . . . 4 (𝑦 = 𝐵 → (𝑢𝑅𝑦𝑢𝑅𝐵))
31, 2bi2anan9 636 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑢𝑅𝐴𝑢𝑅𝐵)))
43exbidv 1917 . 2 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
5 df-coss 38122 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
64, 5brabga 5532 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝑅𝐵 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099   class class class wbr 5145  ccoss 37889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-coss 38122
This theorem is referenced by:  brcoss2  38143  brcoss3  38144  brcosscnvcoss  38145  cocossss  38147  br1cossres  38150  eldmcoss2  38170  brcosscnv  38183  trcoss  38193
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