Theorem br1cossinres 39136

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 6000	. . . 4 ⊢ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
2	1	cosseqi 39116	. . 3 ⊢ ≀ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)
3	2	breqi 5120	. 2 ⊢ (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ 𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶)
4		br1cossres 39128	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶)))
5		brin 5168	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐵 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵))
6		brin 5168	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐶 ↔ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶))
7	5, 6	anbi12i 639	. . . . 5 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)))
8		an2anr 647	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
9	7, 8	bitri 278	. . . 4 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
10	9	rexbii 3119	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
11	4, 10	bitrdi 290	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))
12	3, 11	bitrid 286	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150  ∃wrex 3096   ∩ cin 3912   class class class wbr 5114   ↾ cres 5667   ≀ ccoss 38782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-res 5677  df-coss 39100

This theorem is referenced by:  br1cossinidres  39138  br1cossincnvepres  39139