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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 3965 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆)) | |
2 | df-br 5150 | . 2 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ 𝑆)) | |
3 | df-br 5150 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅) | |
4 | df-br 5150 | . . 3 ⊢ (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆) | |
5 | 3, 4 | anbi12i 625 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆)) |
6 | 1, 2, 5 | 3bitr4i 302 | 1 ⊢ (𝐴(𝑅 ∩ 𝑆)𝐵 ↔ (𝐴𝑅𝐵 ∧ 𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∈ wcel 2104 ∩ cin 3948 ⟨cop 4635 class class class wbr 5149 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3956 df-br 5150 |
This theorem is referenced by: brinxp2 5754 trin2 6125 poirr2 6126 dfpo2 6296 predtrss 6324 tpostpos 8235 erinxp 8789 sbthcl 9099 infxpenlem 10012 fpwwe2lem11 10640 fpwwe2 10642 isinv 17713 isffth2 17873 ffthf1o 17876 ffthoppc 17881 ffthres2c 17897 isunit 20266 opsrtoslem2 21838 posrasymb 32400 trleile 32406 satefvfmla1 34712 brtxp 35154 idsset 35164 dfon3 35166 elfix 35177 dffix2 35179 brcap 35214 funpartlem 35216 trer 35506 fneval 35542 brcnvin 37545 brxrn 37549 brin2 37584 br1cossinres 37622 grumnud 43349 |
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