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Theorem br1cossres 37911
Description: 𝐵 and 𝐶 are cosets by a restriction: a binary relation. (Contributed by Peter Mazsa, 30-Dec-2018.)
Assertion
Ref Expression
br1cossres ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑅𝐶)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossres
StepHypRef Expression
1 brcoss 37903 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑅𝐴)𝐶)))
2 exanres 37767 . 2 ((𝐵𝑉𝐶𝑊) → (∃𝑢(𝑢(𝑅𝐴)𝐵𝑢(𝑅𝐴)𝐶) ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑅𝐶)))
31, 2bitrd 279 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢𝑅𝐵𝑢𝑅𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wex 1774  wcel 2099  wrex 3067   class class class wbr 5148  cres 5680  ccoss 37648
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-res 5690  df-coss 37883
This theorem is referenced by:  br1cossres2  37912  br1cossinres  37919  br1cossxrnres  37920
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