Theorem elrelsrel 38485

Description: The element of the relations class (df-rels 38483) 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 38484	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5648	. 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 2109  Vcvv 3450   ⊆ wss 3917   × cxp 5639  Rel wrel 5646   Rels crels 38178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ss 3934  df-pw 4568  df-rel 5648  df-rels 38483

This theorem is referenced by:  elrelsrelim  38486  elrels5  38487  elrels6  38488  cnvelrels  38493  cosselrels  38494  elrefrelsrel  38518  elcnvrefrelsrel  38534  elsymrelsrel  38555  eltrrelsrel  38579  eleqvrelsrel  38592  elfunsALTVfunALTV  38696  eldisjs5  38725  eldisjsdisj  38726