Theorem br1cossinidres 38431

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 38429	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ ( I ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶))))
2		ideq2 38289	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐵 ↔ 𝑢 = 𝐵))
3	2	elv 3483	. . . . 5 ⊢ (𝑢 I 𝐵 ↔ 𝑢 = 𝐵)
4	3	anbi1i 624	. . . 4 ⊢ ((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝑢 = 𝐵 ∧ 𝑢𝑅𝐵))
5		ideq2 38289	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢 I 𝐶 ↔ 𝑢 = 𝐶))
6	5	elv 3483	. . . . 5 ⊢ (𝑢 I 𝐶 ↔ 𝑢 = 𝐶)
7	6	anbi1i 624	. . . 4 ⊢ ((𝑢 I 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶))
8	4, 7	anbi12i 628	. . 3 ⊢ (((𝑢 I 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 I 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝑢 = 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑢𝑅𝐶)))
9	8	rexbii 3092	. 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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  Vcvv 3478   ∩ cin 3962   class class class wbr 5148   I cid 5582   ↾ cres 5691   ≀ ccoss 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-res 5701  df-coss 38393

This theorem is referenced by: (None)