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Mirrors > Home > MPE Home > Th. List > Mathboxes > brin2 | Structured version Visualization version GIF version |
Description: Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) |
Ref | Expression |
---|---|
brin2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐵〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brin 5131 | . 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) | |
2 | brxrn 36500 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐵〉 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))) | |
3 | 2 | 3anidm23 1420 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐵〉 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵))) |
4 | 1, 3 | bitr4id 290 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 𝐴(𝑅 ⋉ 𝑆)〈𝐵, 𝐵〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2110 ∩ cin 3891 〈cop 4573 class class class wbr 5079 ⋉ cxrn 36328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-fo 6438 df-fv 6440 df-1st 7824 df-2nd 7825 df-xrn 36497 |
This theorem is referenced by: brin3 36532 |
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