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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 35182 | . . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴)) | |
2 | 1 | cosseqi 35203 | . . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴)) |
3 | 2 | breqi 4968 | . 2 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ 〈𝐵, 𝐶〉 ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))〈𝐷, 𝐸〉) |
4 | opex 5248 | . . . 4 ⊢ 〈𝐵, 𝐶〉 ∈ V | |
5 | opex 5248 | . . . 4 ⊢ 〈𝐷, 𝐸〉 ∈ V | |
6 | br1cossres 35215 | . . . 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 688 | . . 3 ⊢ (〈𝐵, 𝐶〉 ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)〈𝐷, 𝐸〉 ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ∧ 𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉)) |
8 | brxrn 35157 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) | |
9 | 8 | el3v1 35024 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)〈𝐵, 𝐶〉 ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶))) |
10 | brxrn 35157 | . . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) | |
11 | 10 | el3v1 35024 | . . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)〈𝐷, 𝐸〉 ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))) |
12 | 9, 11 | bi2anan9 635 | . . . . 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 35030 | . . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))) | |
14 | 12, 13 | syl6bb 288 | . . . 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 3260 | . . 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 284 | . 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 286 | 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 207 ∧ wa 396 ∈ wcel 2081 ∃wrex 3106 Vcvv 3437 〈cop 4478 class class class wbr 4962 ↾ cres 5445 ⋉ cxrn 34984 ≀ ccoss 34985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fo 6231 df-fv 6233 df-1st 7545 df-2nd 7546 df-xrn 35154 df-coss 35190 |
This theorem is referenced by: br1cossxrnidres 35222 br1cossxrncnvepres 35223 br1cossxrncnvssrres 35279 |
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