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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 3758 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) | |
2 | df-br 4584 | . 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑅 ∩ 𝑆)) | |
3 | df-br 4584 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
4 | df-br 4584 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑆) | |
5 | 3, 4 | anbi12i 729 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (〈𝐴, 𝐵〉 ∈ 𝑅 ∧ 〈𝐴, 𝐵〉 ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 291 | 1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∩ cin 3539 〈cop 4131 class class class wbr 4583 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-br 4584 |
This theorem is referenced by: brinxp2 5103 trin2 5438 poirr2 5439 tpostpos 7259 erinxp 7708 sbthcl 7967 infxpenlem 8719 fpwwe2lem12 9342 fpwwe2 9344 isinv 16243 isffth2 16399 ffthf1o 16402 ffthoppc 16407 ffthres2c 16423 isunit 18480 opsrtoslem2 19306 posrasymb 28988 trleile 28997 dfpo2 30898 brtxp 31157 idsset 31167 dfon3 31169 elfix 31180 dffix2 31182 brcap 31217 funpartlem 31219 trer 31480 fneval 31517 |
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