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Theorem br1cossincnvepres 34523
 Description: 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
br1cossincnvepres ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossincnvepres
StepHypRef Expression
1 br1cossinres 34520 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶))))
2 brcnvep 34353 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐵𝐵𝑢))
32elv 34309 . . . . 5 (𝑢 E 𝐵𝐵𝑢)
43anbi1i 733 . . . 4 ((𝑢 E 𝐵𝑢𝑅𝐵) ↔ (𝐵𝑢𝑢𝑅𝐵))
5 brcnvep 34353 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐶𝐶𝑢))
65elv 34309 . . . . 5 (𝑢 E 𝐶𝐶𝑢)
76anbi1i 733 . . . 4 ((𝑢 E 𝐶𝑢𝑅𝐶) ↔ (𝐶𝑢𝑢𝑅𝐶))
84, 7anbi12i 735 . . 3 (((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶)) ↔ ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶)))
98rexbii 3179 . 2 (∃𝑢𝐴 ((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶)) ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶)))
101, 9syl6bb 276 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340   ∩ cin 3714   class class class wbr 4804   E cep 5178  ◡ccnv 5265   ↾ cres 5268   ≀ ccoss 34296 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-xp 5272  df-rel 5273  df-cnv 5274  df-res 5278  df-coss 34492 This theorem is referenced by: (None)
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