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| Mirrors > Home > MPE Home > Th. List > brin | Structured version Visualization version 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 3967 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
| 2 | df-br 5144 | . 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆)) | |
| 3 | df-br 5144 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 4 | df-br 5144 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
| 5 | 3, 4 | anbi12i 628 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
| 6 | 1, 2, 5 | 3bitr4i 303 | 1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∩ cin 3950 〈cop 4632 class class class wbr 5143 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-br 5144 |
| This theorem is referenced by: brinxp2 5763 trin2 6143 poirr2 6144 dfpo2 6316 predtrss 6343 tpostpos 8271 brinxper 8774 erinxp 8831 sbthcl 9135 infxpenlem 10053 fpwwe2lem11 10681 fpwwe2 10683 isinv 17804 isffth2 17963 ffthf1o 17966 ffthoppc 17971 ffthres2c 17987 isunit 20373 opsrtoslem2 22080 posrasymb 32955 trleile 32961 satefvfmla1 35430 brtxp 35881 idsset 35891 dfon3 35893 elfix 35904 dffix2 35906 brcap 35941 funpartlem 35943 trer 36317 fneval 36353 brcnvin 38371 brxrn 38375 brin2 38410 br1cossinres 38448 grumnud 44305 |
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