Theorem br1cossinidres 37915

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 37913	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶))))
2		ideq2 37773	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐵 ↔ 𝑢 = 𝐵))
3	2	elv 3476	. . . . 5 ⊢ (𝑢 I 𝐵 ↔ 𝑢 = 𝐵)
4	3	anbi1i 623	. . . 4 ⊢ ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐵 ∧ 𝑢𝑅𝐵))
5		ideq2 37773	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
6	5	elv 3476	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
7	6	anbi1i 623	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 627	. . 3 ⊢ (((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3090	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))
10	1, 9	bitrdi 287	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3066  Vcvv 3470   ∩ cin 3944   class class class wbr 5142   I cid 5569   ↾ cres 5674   ≀ ccoss 37642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-res 5684  df-coss 37877

This theorem is referenced by: (None)