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Mirrors > Home > ILE Home > Th. List > brin | GIF version |
Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.) |
Ref | Expression |
---|---|
brin | ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3254 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 3925 | . 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆)) | |
3 | df-br 3925 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 3925 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | anbi12i 455 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 211 | 1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∩ cin 3065 〈cop 3525 class class class wbr 3924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-in 3072 df-br 3925 |
This theorem is referenced by: brinxp2 4601 trin2 4925 poirr2 4926 cnvin 4941 tpostpos 6154 erinxp 6496 |
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