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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 range Cartesian product 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 38404 | . . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | |
| 2 | 1 | cosseqi 38428 | . . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴)) | 
| 3 | 2 | breqi 5149 | . 2 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉) | 
| 4 | opex 5469 | . . . 4 ⊢ 〈𝐵, 𝐶〉 ∈ V | |
| 5 | opex 5469 | . . . 4 ⊢ 〈𝐷, 𝐸〉 ∈ V | |
| 6 | br1cossres 38440 | . . . 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 38375 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | |
| 9 | 8 | el3v1 38225 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | 
| 10 | brxrn 38375 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) | |
| 11 | 10 | el3v1 38225 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) | 
| 12 | 9, 11 | bi2anan9 638 | . . . . 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 636 | . . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))) | |
| 14 | 12, 13 | bitrdi 287 | . . . 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 3179 | . . 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 | bitrid 283 | . 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 285 | 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 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 〈cop 4632 class class class wbr 5143 ↾ cres 5687 ⋉ cxrn 38181 ≀ ccoss 38182 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-xrn 38372 df-coss 38412 | 
| This theorem is referenced by: br1cossxrnidres 38452 br1cossxrncnvepres 38453 br1cossxrncnvssrres 38509 | 
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