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Theorem br1cnvssrres 37013
Description: Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)
Assertion
Ref Expression
br1cnvssrres (𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))

Proof of Theorem br1cnvssrres
StepHypRef Expression
1 relres 5967 . . 3 Rel ( S ↾ 𝐴)
21relbrcnv 6060 . 2 (𝐵( S ↾ 𝐴)𝐶𝐶( S ↾ 𝐴)𝐵)
3 brssrres 37012 . 2 (𝐵𝑉 → (𝐶( S ↾ 𝐴)𝐵 ↔ (𝐶𝐴𝐶𝐵)))
42, 3bitrid 283 1 (𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wss 3911   class class class wbr 5106  ccnv 5633  cres 5636   S cssr 36683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-res 5646  df-ssr 37006
This theorem is referenced by: (None)
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