Theorem elrelsrel 38809

Description: The element of the relations class (df-rels 38807) 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 38808	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5625	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 290	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 207   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883   × cxp 5616  Rel wrel 5623   Rels crels 38552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ss 3900  df-pw 4531  df-rel 5625  df-rels 38807

This theorem is referenced by:  elrelsrelim  38810  elrels5  38811  elrels6  38812  cosselrels  38942  cnvelrels  38943  elrefrelsrel  38967  elcnvrefrelsrel  38983  elsymrelsrel  39008  eltrrelsrel  39032  eleqvrelsrel  39045  elfunsALTVfunALTV  39149  eldisjs5  39190  eldisjsdisj  39191