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Theorem br1cossinres 35681
 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 5865 . . . 4 (𝑅 ∩ (𝑆𝐴)) = ((𝑅𝑆) ↾ 𝐴)
21cosseqi 35666 . . 3 ≀ (𝑅 ∩ (𝑆𝐴)) = ≀ ((𝑅𝑆) ↾ 𝐴)
32breqi 5064 . 2 (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶)
4 br1cossres 35678 . . 3 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢𝐴 (𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶)))
5 brin 5110 . . . . . 6 (𝑢(𝑅𝑆)𝐵 ↔ (𝑢𝑅𝐵𝑢𝑆𝐵))
6 brin 5110 . . . . . 6 (𝑢(𝑅𝑆)𝐶 ↔ (𝑢𝑅𝐶𝑢𝑆𝐶))
75, 6anbi12i 628 . . . . 5 ((𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ((𝑢𝑅𝐵𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶𝑢𝑆𝐶)))
8 an2anr 35492 . . . . 5 (((𝑢𝑅𝐵𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶𝑢𝑆𝐶)) ↔ ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
97, 8bitri 277 . . . 4 ((𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
109rexbii 3247 . . 3 (∃𝑢𝐴 (𝑢(𝑅𝑆)𝐵𝑢(𝑅𝑆)𝐶) ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶)))
114, 10syl6bb 289 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ ((𝑅𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))
123, 11syl5bb 285 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢𝑆𝐵𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶𝑢𝑅𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  ∃wrex 3139   ∩ cin 3934   class class class wbr 5058   ↾ cres 5551   ≀ ccoss 35447 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-res 5561  df-coss 35653 This theorem is referenced by:  br1cossinidres  35683  br1cossincnvepres  35684
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