Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrelsrel Structured version   Visualization version   GIF version

Theorem elrelsrel 36995
Description: The element of the relations class (df-rels 36993) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.)
Assertion
Ref Expression
elrelsrel (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Proof of Theorem elrelsrel
StepHypRef Expression
1 elrels2 36994 . 2 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2 df-rel 5641 . 2 (Rel 𝑅𝑅 ⊆ (V × V))
31, 2bitr4di 289 1 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  Vcvv 3444  wss 3911   × cxp 5632  Rel wrel 5639   Rels crels 36682
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
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563  df-rel 5641  df-rels 36993
This theorem is referenced by:  elrelsrelim  36996  elrels5  36997  elrels6  36998  cnvelrels  37003  cosselrels  37004  elrefrelsrel  37028  elcnvrefrelsrel  37044  elsymrelsrel  37065  eltrrelsrel  37089  eleqvrelsrel  37102  elfunsALTVfunALTV  37205  eldisjs5  37234  eldisjsdisj  37235
  Copyright terms: Public domain W3C validator