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Theorem br1cnvssrres 34578
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 5584 . . 3 Rel ( S ↾ 𝐴)
21relbrcnv 5664 . 2 (𝐵( S ↾ 𝐴)𝐶𝐶( S ↾ 𝐴)𝐵)
3 brssrres 34577 . 2 (𝐵𝑉 → (𝐶( S ↾ 𝐴)𝐵 ↔ (𝐶𝐴𝐶𝐵)))
42, 3syl5bb 272 1 (𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wss 3715   class class class wbr 4804  ccnv 5265  cres 5268   S cssr 34299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-res 5278  df-ssr 34571
This theorem is referenced by: (None)
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