Theorem elrelsrel 38902

Description: The element of the relations class (df-rels 38900) 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 38901	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5650	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 291	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902   × cxp 5641  Rel wrel 5648   Rels crels 38645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ss 3919  df-pw 4554  df-rel 5650  df-rels 38900

This theorem is referenced by:  elrelsrelim  38903  elrels5  38904  elrels6  38905  cosselrels  39035  cnvelrels  39036  elrefrelsrel  39060  elcnvrefrelsrel  39076  elsymrelsrel  39101  eltrrelsrel  39125  eleqvrelsrel  39138  elfunsALTVfunALTV  39242  eldisjs5  39283  eldisjsdisj  39284