Theorem br1cossinres 38433

Description: 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
br1cossinres	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))

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

Proof of Theorem br1cossinres

Step	Hyp	Ref	Expression
1		inres 5970	. . . 4 ⊢ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
2	1	cosseqi 38413	. . 3 ⊢ ≀ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)
3	2	breqi 5115	. 2 ⊢ (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ 𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶)
4		br1cossres 38425	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶)))
5		brin 5161	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐵 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵))
6		brin 5161	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐶 ↔ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶))
7	5, 6	anbi12i 628	. . . . 5 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)))
8		an2anr 636	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
9	7, 8	bitri 275	. . . 4 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
10	9	rexbii 3077	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
11	4, 10	bitrdi 287	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))
12	3, 11	bitrid 283	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∃wrex 3054   ∩ cin 3915   class class class wbr 5109   ↾ cres 5642   ≀ ccoss 38164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-xp 5646  df-res 5652  df-coss 38397

This theorem is referenced by:  br1cossinidres  38435  br1cossincnvepres  38436