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Theorem elrelsrel 38488
Description: The element of the relations class (df-rels 38486) 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 38487 . 2 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2 df-rel 5692 . 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 206  wcel 2108  Vcvv 3480  wss 3951   × cxp 5683  Rel wrel 5690   Rels crels 38184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-pw 4602  df-rel 5692  df-rels 38486
This theorem is referenced by:  elrelsrelim  38489  elrels5  38490  elrels6  38491  cnvelrels  38496  cosselrels  38497  elrefrelsrel  38521  elcnvrefrelsrel  38537  elsymrelsrel  38558  eltrrelsrel  38582  eleqvrelsrel  38595  elfunsALTVfunALTV  38698  eldisjs5  38727  eldisjsdisj  38728
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