Theorem br1cossinres 36534

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 5903	. . . 4 ⊢ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ 𝑆) ↾ 𝐴)
2	1	cosseqi 36519	. . 3 ⊢ ≀ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)
3	2	breqi 5081	. 2 ⊢ (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ 𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶)
4		br1cossres 36531	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶)))
5		brin 5127	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐵 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵))
6		brin 5127	. . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐶 ↔ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶))
7	5, 6	anbi12i 626	. . . . 5 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)))
8		an2anr 36347	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
9	7, 8	bitri 274	. . . 4 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
10	9	rexbii 3176	. . 3 ⊢ (∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))
11	4, 10	bitrdi 286	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))
12	3, 11	syl5bb 282	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2107  ∃wrex 3063   ∩ cin 3887   class class class wbr 5075   ↾ cres 5587   ≀ ccoss 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3429  df-dif 3891  df-un 3893  df-in 3895  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5076  df-opab 5138  df-xp 5591  df-res 5597  df-coss 36506

This theorem is referenced by:  br1cossinidres  36536  br1cossincnvepres  36537