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Theorem br1cossincnvepres 38432
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 38429 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶))))
2 brcnvep 38247 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐵𝐵𝑢))
32elv 3483 . . . . 5 (𝑢 E 𝐵𝐵𝑢)
43anbi1i 624 . . . 4 ((𝑢 E 𝐵𝑢𝑅𝐵) ↔ (𝐵𝑢𝑢𝑅𝐵))
5 brcnvep 38247 . . . . . 6 (𝑢 ∈ V → (𝑢 E 𝐶𝐶𝑢))
65elv 3483 . . . . 5 (𝑢 E 𝐶𝐶𝑢)
76anbi1i 624 . . . 4 ((𝑢 E 𝐶𝑢𝑅𝐶) ↔ (𝐶𝑢𝑢𝑅𝐶))
84, 7anbi12i 628 . . 3 (((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶)) ↔ ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶)))
98rexbii 3092 . 2 (∃𝑢𝐴 ((𝑢 E 𝐵𝑢𝑅𝐵) ∧ (𝑢 E 𝐶𝑢𝑅𝐶)) ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶)))
101, 9bitrdi 287 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( E ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝐵𝑢𝑢𝑅𝐵) ∧ (𝐶𝑢𝑢𝑅𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wrex 3068  Vcvv 3478  cin 3962   class class class wbr 5148   E cep 5588  ccnv 5688  cres 5691  ccoss 38162
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-res 5701  df-coss 38393
This theorem is referenced by: (None)
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