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Theorem eqres 37673
Description: Converting a class constant definition by restriction (like df-ers 37997 or df-parts 38099) 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 5988 . 2 (𝐵𝑉 → (𝐴(𝑆𝐶)𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
42, 3bitrid 283 1 (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105   class class class wbr 5148  cres 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688
This theorem is referenced by:  brers  38001  brparts  38105
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