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Theorem brin2 35656
Description: Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.)
Assertion
Ref Expression
brin2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))

Proof of Theorem brin2
StepHypRef Expression
1 brxrn 35625 . . 3 ((𝐴𝑉𝐵𝑊𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
213anidm23 1417 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
3 brin 5117 . 2 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
42, 3syl6rbbr 292 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2110  cin 3934  cop 4572   class class class wbr 5065  cxrn 35451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-1st 7688  df-2nd 7689  df-xrn 35622
This theorem is referenced by:  brin3  35657
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