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Theorem br1cossinidres 38451
Description: 𝐵 and 𝐶 are cosets by an intersection with the restricted identity class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
br1cossinidres ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossinidres
StepHypRef Expression
1 br1cossinres 38449 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶))))
2 ideq2 38309 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐵𝑢 = 𝐵))
32elv 3484 . . . . 5 (𝑢 I 𝐵𝑢 = 𝐵)
43anbi1i 624 . . . 4 ((𝑢 I 𝐵𝑢𝑅𝐵) ↔ (𝑢 = 𝐵𝑢𝑅𝐵))
5 ideq2 38309 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐶𝑢 = 𝐶))
65elv 3484 . . . . 5 (𝑢 I 𝐶𝑢 = 𝐶)
76anbi1i 624 . . . 4 ((𝑢 I 𝐶𝑢𝑅𝐶) ↔ (𝑢 = 𝐶𝑢𝑅𝐶))
84, 7anbi12i 628 . . 3 (((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
98rexbii 3093 . 2 (∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
101, 9bitrdi 287 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wrex 3069  Vcvv 3479  cin 3949   class class class wbr 5142   I cid 5576  cres 5686  ccoss 38183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-res 5696  df-coss 38413
This theorem is referenced by: (None)
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