Theorem br1cossinidres 39663

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 39661	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶))))
2		ideq2 39437	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐵 ↔ 𝑢 = 𝐵))
3	2	elv 3459	. . . . 5 ⊢ (𝑢 I 𝐵 ↔ 𝑢 = 𝐵)
4	3	anbi1i 636	. . . 4 ⊢ ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐵 ∧ 𝑢𝑅𝐵))
5		ideq2 39437	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
6	5	elv 3459	. . . . 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 3113	. 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 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ∩ cin 4092   class class class wbr 5106   I cid 5549   ↾ cres 5657   ≀ ccoss 39307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667  df-coss 39625

This theorem is used by: (None)