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Theorem eqres 38322
Description: Converting a class constant definition by restriction (like df-ers 38645 or df-parts 38747) into a binary relation. (Contributed by Peter Mazsa, 1-Oct-2018.)
Hypothesis
Ref Expression
eqres.1 𝑅 = (𝑆𝐶)
Assertion
Ref Expression
eqres (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))

Proof of Theorem eqres
StepHypRef Expression
1 eqres.1 . . 3 𝑅 = (𝑆𝐶)
21breqi 5154 . 2 (𝐴𝑅𝐵𝐴(𝑆𝐶)𝐵)
3 brres 6007 . 2 (𝐵𝑉 → (𝐴(𝑆𝐶)𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
42, 3bitrid 283 1 (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  cres 5691
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  brers  38649  brparts  38753
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