Theorem elrelsrel 38627

Description: The element of the relations class (df-rels 38625) 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 38626	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5631	. 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 3440   ⊆ wss 3901   × cxp 5622  Rel wrel 5629   Rels crels 38385

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ss 3918  df-pw 4556  df-rel 5631  df-rels 38625

This theorem is referenced by:  elrelsrelim  38628  elrels5  38629  elrels6  38630  cosselrels  38748  cnvelrels  38749  elrefrelsrel  38773  elcnvrefrelsrel  38789  elsymrelsrel  38814  eltrrelsrel  38838  eleqvrelsrel  38851  elfunsALTVfunALTV  38956  eldisjs5  38985  eldisjsdisj  38986