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Theorem br1cnvssrres 34749
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 5636 . . 3 Rel ( S ↾ 𝐴)
21relbrcnv 5723 . 2 (𝐵( S ↾ 𝐴)𝐶𝐶( S ↾ 𝐴)𝐵)
3 brssrres 34748 . 2 (𝐵𝑉 → (𝐶( S ↾ 𝐴)𝐵 ↔ (𝐶𝐴𝐶𝐵)))
42, 3syl5bb 275 1 (𝐵𝑉 → (𝐵( S ↾ 𝐴)𝐶 ↔ (𝐶𝐴𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wss 3769   class class class wbr 4843  ccnv 5311  cres 5314   S cssr 34472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-res 5324  df-ssr 34742
This theorem is referenced by: (None)
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