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Theorem br1cossinidres 36331
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 36329 . 2 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶))))
2 ideq2 36207 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐵𝑢 = 𝐵))
32elv 3427 . . . . 5 (𝑢 I 𝐵𝑢 = 𝐵)
43anbi1i 627 . . . 4 ((𝑢 I 𝐵𝑢𝑅𝐵) ↔ (𝑢 = 𝐵𝑢𝑅𝐵))
5 ideq2 36207 . . . . . 6 (𝑢 ∈ V → (𝑢 I 𝐶𝑢 = 𝐶))
65elv 3427 . . . . 5 (𝑢 I 𝐶𝑢 = 𝐶)
76anbi1i 627 . . . 4 ((𝑢 I 𝐶𝑢𝑅𝐶) ↔ (𝑢 = 𝐶𝑢𝑅𝐶))
84, 7anbi12i 630 . . 3 (((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
98rexbii 3176 . 2 (∃𝑢𝐴 ((𝑢 I 𝐵𝑢𝑅𝐵) ∧ (𝑢 I 𝐶𝑢𝑅𝐶)) ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶)))
101, 9bitrdi 290 1 ((𝐵𝑉𝐶𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢𝐴 ((𝑢 = 𝐵𝑢𝑅𝐵) ∧ (𝑢 = 𝐶𝑢𝑅𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2111  wrex 3063  Vcvv 3421  cin 3880   class class class wbr 5068   I cid 5469  cres 5568  ccoss 36097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-res 5578  df-coss 36301
This theorem is referenced by: (None)
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