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Theorem eqres 36098
Description: Converting a class constant definition by restriction (like df-ers 36398 or ~? df-parts ) 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 5036 . 2 (𝐴𝑅𝐵𝐴(𝑆𝐶)𝐵)
3 brres 5832 . 2 (𝐵𝑉 → (𝐴(𝑆𝐶)𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
42, 3syl5bb 286 1 (𝐵𝑉 → (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐴𝑆𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114   class class class wbr 5030  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-res 5537
This theorem is referenced by:  brers  36402
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