Theorem elrelsrel 39354

Description: The element of the relations class (df-rels 39352) 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 39353	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5658	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 292	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899   × cxp 5649  Rel wrel 5656   Rels crels 39097

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ss 3916  df-pw 4559  df-rel 5658  df-rels 39352

This theorem is used by:  elrelsrelim  39355  elrels5  39356  elrels6  39357  cosselrels  39487  cnvelrels  39488  elrefrelsrel  39512  elcnvrefrelsrel  39528  elsymrelsrel  39553  eltrrelsrel  39577  eleqvrelsrel  39590  elfunsALTVfunALTV  39694  eldisjs5  39735  eldisjsdisj  39736