Theorem elrelsrel 36601

Description: The element of the relations class (df-rels 36599) 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 36600	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5597	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 289	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2110  Vcvv 3431   ⊆ wss 3892   × cxp 5588  Rel wrel 5595   Rels crels 36331

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909  df-pw 4541  df-rel 5597  df-rels 36599

This theorem is referenced by:  elrelsrelim  36602  elrels5  36603  elrels6  36604  cnvelrels  36609  cosselrels  36610  elrefrelsrel  36633  elcnvrefrelsrel  36646  elsymrelsrel  36667  eltrrelsrel  36691  eleqvrelsrel  36703  elfunsALTVfunALTV  36804  eldisjs5  36833  eldisjsdisj  36834