Theorem elrelsrel 38489

Description: The element of the relations class (df-rels 38487) 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 38488	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5628	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 289	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  Vcvv 3437   ⊆ wss 3898   × cxp 5619  Rel wrel 5626   Rels crels 38247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ss 3915  df-pw 4553  df-rel 5628  df-rels 38487

This theorem is referenced by:  elrelsrelim  38490  elrels5  38491  elrels6  38492  cosselrels  38610  cnvelrels  38611  elrefrelsrel  38635  elcnvrefrelsrel  38651  elsymrelsrel  38676  eltrrelsrel  38700  eleqvrelsrel  38713  elfunsALTVfunALTV  38818  eldisjs5  38847  eldisjsdisj  38848