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Theorem br1cossinres 36565
Description: 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
br1cossinres ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑆   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossinres
StepHypRef Expression
1 inres 5909 . . . 4 (𝑅 ∩ (𝑆𝐴)) = ((𝑅𝑆) ↾ 𝐴)
21cosseqi 36550 . . 3 ≀ (𝑅 ∩ (𝑆𝐴)) = ≀ ((𝑅𝑆) ↾ 𝐴)
32breqi 5080 . 2 (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶)
4 br1cossres 36562 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶)))
5 brin 5126 . . . . . 6 (𝑢(𝑅𝑆)𝐵 ↔ (𝑢𝑅𝐵𝑢𝑆𝐵))
6 brin 5126 . . . . . 6 (𝑢(𝑅𝑆)𝐶 ↔ (𝑢𝑅𝐶𝑢𝑆𝐶))
75, 6anbi12i 627 . . . . 5 ((𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ((𝑢𝑅𝐵𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶𝑢𝑆𝐶)))
8 an2anr 36378 . . . . 5 (((𝑢𝑅𝐵𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶𝑢𝑆𝐶)) ↔ ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
97, 8bitri 274 . . . 4 ((𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
109rexbii 3181 . . 3 (∃𝑢𝐴 (𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
114, 10bitrdi 287 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))
123, 11syl5bb 283 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wrex 3065  cin 3886   class class class wbr 5074  cres 5591  ccoss 36333
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-res 5601  df-coss 36537
This theorem is referenced by:  br1cossinidres  36567  br1cossincnvepres  36568
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