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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossxrnres | Structured version Visualization version GIF version |
Description: 〈𝐵, 𝐶〉 and 〈𝐷, 𝐸〉 are cosets by an intersection with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.) |
Ref | Expression |
---|---|
br1cossxrnres | ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnres2 36266 | . . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | |
2 | 1 | cosseqi 36287 | . . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴)) |
3 | 2 | breqi 5059 | . 2 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉) |
4 | opex 5348 | . . . 4 ⊢ 〈𝐵, 𝐶〉 ∈ V | |
5 | opex 5348 | . . . 4 ⊢ 〈𝐷, 𝐸〉 ∈ V | |
6 | br1cossres 36299 | . . . 4 ⊢ ((〈𝐵, 𝐶〉 ∈ V ∧ 〈𝐷, 𝐸〉 ∈ V) → (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉))) | |
7 | 4, 5, 6 | mp2an 692 | . . 3 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉)) |
8 | brxrn 36241 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | |
9 | 8 | el3v1 36109 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) |
10 | brxrn 36241 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) | |
11 | 10 | el3v1 36109 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) |
12 | 9, 11 | bi2anan9 639 | . . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → ((𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))) |
13 | an2anr 36115 | . . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))) | |
14 | 12, 13 | bitrdi 290 | . . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → ((𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) |
15 | 14 | rexbidv 3216 | . . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉) ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) |
16 | 7, 15 | syl5bb 286 | . 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) |
17 | 3, 16 | bitr3id 288 | 1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∃wrex 3062 Vcvv 3408 〈cop 4547 class class class wbr 5053 ↾ cres 5553 ⋉ cxrn 36069 ≀ ccoss 36070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-fo 6386 df-fv 6388 df-1st 7761 df-2nd 7762 df-xrn 36238 df-coss 36274 |
This theorem is referenced by: br1cossxrnidres 36306 br1cossxrncnvepres 36307 br1cossxrncnvssrres 36363 |
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