Theorem br1cossinidres 39547

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	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝑅   𝑢,𝑉   𝑢,𝑊

Proof of Theorem br1cossinidres

Step	Hyp	Ref	Expression
1		br1cossinres 39545	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶))))
2		ideq2 39321	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐵 ↔ 𝑢 = 𝐵))
3	2	elv 3455	. . . . 5 ⊢ (𝑢 I 𝐵 ↔ 𝑢 = 𝐵)
4	3	anbi1i 636	. . . 4 ⊢ ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐵 ∧ 𝑢𝑅𝐵))
5		ideq2 39321	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
6	5	elv 3455	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
7	6	anbi1i 636	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 640	. . 3 ⊢ (((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3109	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))
10	1, 9	bitrdi 290	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ∩ cin 4089   class class class wbr 5102   I cid 5544   ↾ cres 5652   ≀ ccoss 39191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-res 5662  df-coss 39509

This theorem is used by: (None)