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Theorem brssrres 38486
Description: Restricted subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
Assertion
Ref Expression
brssrres (𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))

Proof of Theorem brssrres
StepHypRef Expression
1 brres 6007 . 2 (𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 S 𝐶)))
2 brssr 38483 . . 3 (𝐶𝑉 → (𝐵 S 𝐶𝐵𝐶))
32anbi2d 630 . 2 (𝐶𝑉 → ((𝐵𝐴𝐵 S 𝐶) ↔ (𝐵𝐴𝐵𝐶)))
41, 3bitrd 279 1 (𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wss 3963   class class class wbr 5148  cres 5691   S cssr 38165
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-rel 5696  df-res 5701  df-ssr 38480
This theorem is referenced by:  br1cnvssrres  38487
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