![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossinres | Structured version Visualization version GIF version |
Description: 𝐵 and 𝐶 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
br1cossinres | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐶 ∧ 𝑢𝑅𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inres 6029 | . . . 4 ⊢ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ((𝑅 ∩ 𝑆) ↾ 𝐴) | |
2 | 1 | cosseqi 38385 | . . 3 ⊢ ≀ (𝑅 ∩ (𝑆 ↾ 𝐴)) = ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴) |
3 | 2 | breqi 5172 | . 2 ⊢ (𝐵 ≀ (𝑅 ∩ (𝑆 ↾ 𝐴))𝐶 ↔ 𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶) |
4 | br1cossres 38397 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ ((𝑅 ∩ 𝑆) ↾ 𝐴)𝐶 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶))) | |
5 | brin 5218 | . . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐵 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵)) | |
6 | brin 5218 | . . . . . 6 ⊢ (𝑢(𝑅 ∩ 𝑆)𝐶 ↔ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶)) | |
7 | 5, 6 | anbi12i 627 | . . . . 5 ⊢ ((𝑢(𝑅 ∩ 𝑆)𝐵 ∧ 𝑢(𝑅 ∩ 𝑆)𝐶) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐵) ∧ (𝑢𝑅𝐶 ∧ 𝑢𝑆𝐶))) |
8 | an2anr 635 | . . . . 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 3100 | . . 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)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3076 ∩ cin 3975 class class class wbr 5166 ↾ cres 5702 ≀ ccoss 38137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-xp 5706 df-res 5712 df-coss 38369 |
This theorem is referenced by: br1cossinidres 38407 br1cossincnvepres 38408 |
Copyright terms: Public domain | W3C validator |