Theorem elrelsrel 35887

Description: The element of the relations class (df-rels 35885) 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 35886	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5526	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	syl6bbr 292	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881   × cxp 5517  Rel wrel 5524   Rels crels 35615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-rel 5526  df-rels 35885

This theorem is referenced by:  elrelsrelim  35888  elrels5  35889  elrels6  35890  cnvelrels  35895  cosselrels  35896  elrefrelsrel  35919  elcnvrefrelsrel  35932  elsymrelsrel  35953  eltrrelsrel  35977  eleqvrelsrel  35989  elfunsALTVfunALTV  36090  eldisjs5  36119  eldisjsdisj  36120