Theorem elrelsrel 38690

Description: The element of the relations class (df-rels 38688) 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 38689	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5639	. 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 2114  Vcvv 3442   ⊆ wss 3903   × cxp 5630  Rel wrel 5637   Rels crels 38433

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-pw 4558  df-rel 5639  df-rels 38688

This theorem is referenced by:  elrelsrelim  38691  elrels5  38692  elrels6  38693  cosselrels  38823  cnvelrels  38824  elrefrelsrel  38848  elcnvrefrelsrel  38864  elsymrelsrel  38889  eltrrelsrel  38913  eleqvrelsrel  38926  elfunsALTVfunALTV  39030  eldisjs5  39071  eldisjsdisj  39072