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Theorem br1cossinidres 35569
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 35567 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶))))
2 ideq2 35446 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐵𝑢 = 𝐵))
32elv 3497 . . . . 5 (𝑢 I 𝐵𝑢 = 𝐵)
43anbi1i 623 . . . 4 ((𝑢 I 𝐵𝑢𝑅𝐵) ↔ (𝑢 = 𝐵𝑢𝑅𝐵))
5 ideq2 35446 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐶𝑢 = 𝐶))
65elv 3497 . . . . 5 (𝑢 I 𝐶𝑢 = 𝐶)
76anbi1i 623 . . . 4 ((𝑢 I 𝐶𝑢𝑅𝐶) ↔ (𝑢 = 𝐶𝑢𝑅𝐶))
84, 7anbi12i 626 . . 3 (((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
98rexbii 3244 . 2 (∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
101, 9syl6bb 288 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  cin 3932   class class class wbr 5057   I cid 5452  cres 5550  ccoss 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-res 5560  df-coss 35539
This theorem is referenced by: (None)
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