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Theorem brin 4856
Description: The intersection of two relations. (Contributed by FL, 7-Oct-2008.)
Assertion
Ref Expression
brin (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))

Proof of Theorem brin
StepHypRef Expression
1 elin 3939 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
2 df-br 4805 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑆))
3 df-br 4805 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
4 df-br 4805 . . 3 (𝐴𝑆𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑆)
53, 4anbi12i 735 . 2 ((𝐴𝑅𝐵𝐴𝑆𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝑆))
61, 2, 53bitr4i 292 1 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2139  cin 3714  cop 4327   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-br 4805
This theorem is referenced by:  brinxp2  5337  trin2  5677  poirr2  5678  tpostpos  7541  erinxp  7988  sbthcl  8247  infxpenlem  9026  fpwwe2lem12  9655  fpwwe2  9657  isinv  16621  isffth2  16777  ffthf1o  16780  ffthoppc  16785  ffthres2c  16801  isunit  18857  opsrtoslem2  19687  posrasymb  29966  trleile  29975  dfpo2  31952  brtxp  32293  idsset  32303  dfon3  32305  elfix  32316  dffix2  32318  brcap  32353  funpartlem  32355  trer  32616  fneval  32653  brinxp2ALTV  34358  brxrn  34459  brin2  34490  br1cossinres  34520
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