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Theorem brin3 36273
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 36272 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))
2 opidg 4803 . . . 4 (𝐵𝑊 → ⟨𝐵, 𝐵⟩ = {{𝐵}})
32adantl 485 . . 3 ((𝐴𝑉𝐵𝑊) → ⟨𝐵, 𝐵⟩ = {{𝐵}})
43breq2d 5065 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ 𝐴(𝑅𝑆){{𝐵}}))
51, 4bitrd 282 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆){{𝐵}}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  cin 3865  {csn 4541  cop 4547   class class class wbr 5053  cxrn 36069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-1st 7761  df-2nd 7762  df-xrn 36238
This theorem is referenced by: (None)
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