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Theorem brssrres 35849
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 5847 . 2 (𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 S 𝐶)))
2 brssr 35846 . . 3 (𝐶𝑉 → (𝐵 S 𝐶𝐵𝐶))
32anbi2d 631 . 2 (𝐶𝑉 → ((𝐵𝐴𝐵 S 𝐶) ↔ (𝐵𝐴𝐵𝐶)))
41, 3bitrd 282 1 (𝐶𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wss 3919   class class class wbr 5052  cres 5544   S cssr 35561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-res 5554  df-ssr 35843
This theorem is referenced by:  br1cnvssrres  35850
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