Theorem elrelsrel 38488

Description: The element of the relations class (df-rels 38486) 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 38487	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5692	. 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 2108  Vcvv 3480   ⊆ wss 3951   × cxp 5683  Rel wrel 5690   Rels crels 38184

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3968  df-pw 4602  df-rel 5692  df-rels 38486

This theorem is referenced by:  elrelsrelim  38489  elrels5  38490  elrels6  38491  cnvelrels  38496  cosselrels  38497  elrefrelsrel  38521  elcnvrefrelsrel  38537  elsymrelsrel  38558  eltrrelsrel  38582  eleqvrelsrel  38595  elfunsALTVfunALTV  38698  eldisjs5  38727  eldisjsdisj  38728