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Theorem brin2 34509
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 34478 . . 3 ((𝐴𝑉𝐵𝑊𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
213anidm23 1531 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩ ↔ (𝐴𝑅𝐵𝐴𝑆𝐵)))
3 brin 4838 . 2 (𝐴(𝑅𝑆)𝐵 ↔ (𝐴𝑅𝐵𝐴𝑆𝐵))
42, 3syl6rbbr 279 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝐵, 𝐵⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  cin 3722  cop 4322   class class class wbr 4786  cxrn 34314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-xrn 34475
This theorem is referenced by:  brin3  34510
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