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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 35653 | . . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | |
2 | 1 | cosseqi 35674 | . . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴)) |
3 | 2 | breqi 5074 | . 2 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉) |
4 | opex 5358 | . . . 4 ⊢ 〈𝐵, 𝐶〉 ∈ V | |
5 | opex 5358 | . . . 4 ⊢ 〈𝐷, 𝐸〉 ∈ V | |
6 | br1cossres 35686 | . . . 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 690 | . . 3 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉)) |
8 | brxrn 35628 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | |
9 | 8 | el3v1 35494 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) |
10 | brxrn 35628 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) | |
11 | 10 | el3v1 35494 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) |
12 | 9, 11 | bi2anan9 637 | . . . . 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 35500 | . . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))) | |
14 | 12, 13 | syl6bb 289 | . . . 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 3299 | . . 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 285 | . 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 | syl5bbr 287 | 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 208 ∧ wa 398 ∈ wcel 2114 ∃wrex 3141 Vcvv 3496 〈cop 4575 class class class wbr 5068 ↾ cres 5559 ⋉ cxrn 35454 ≀ ccoss 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-xrn 35625 df-coss 35661 |
This theorem is referenced by: br1cossxrnidres 35693 br1cossxrncnvepres 35694 br1cossxrncnvssrres 35750 |
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