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Theorem brin3 36803
Description: Binary relation on an intersection is a special case of binary relation on range Cartesian product. (Contributed by Peter Mazsa, 21-Aug-2021.) (Avoid depending on this detail.)
Assertion
Ref Expression
brin3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆){{𝐵}}))

Proof of Theorem brin3
StepHypRef Expression
1 brin2 36802 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))
2 opidg 4847 . . . 4 (𝐵𝑊 → ⟨𝐵, 𝐵⟩ = {{𝐵}})
32adantl 482 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐵, 𝐵⟩ = {{𝐵}})
43breq2d 5115 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ 𝐴(𝑅𝑆){{𝐵}}))
51, 4bitrd 278 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆){{𝐵}}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cin 3907  {csn 4584  cop 4590   class class class wbr 5103  cxrn 36564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7913  df-2nd 7914  df-xrn 36764
This theorem is referenced by: (None)
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