Theorem elrelsrel 38976

Description: The element of the relations class (df-rels 38974) and the relation predicate are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 24-Nov-2018.)

Assertion

Ref	Expression
elrelsrel	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Proof of Theorem elrelsrel

Step	Hyp	Ref	Expression
1		elrels2 38975	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5666	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 292	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913   × cxp 5657  Rel wrel 5664   Rels crels 38719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ss 3930  df-pw 4566  df-rel 5666  df-rels 38974

This theorem is referenced by:  elrelsrelim  38977  elrels5  38978  elrels6  38979  cosselrels  39109  cnvelrels  39110  elrefrelsrel  39134  elcnvrefrelsrel  39150  elsymrelsrel  39175  eltrrelsrel  39199  eleqvrelsrel  39212  elfunsALTVfunALTV  39316  eldisjs5  39357  eldisjsdisj  39358