Theorem br1cossinres 38438

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 5968	. . . 4 ⊢ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
2	1	cosseqi 38418	. . 3 ⊢ ≀ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)
3	2	breqi 5113	. 2 ⊢ (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ 𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶)
4		br1cossres 38430	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶)))
5		brin 5159	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐵 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵))
6		brin 5159	. . . . . 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 3076	. . 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 3053   ∩ cin 3913   class class class wbr 5107   ↾ cres 5640   ≀ ccoss 38169

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-res 5650  df-coss 38402

This theorem is referenced by:  br1cossinidres  38440  br1cossincnvepres  38441