Theorem elrelsrel 38505

Description: The element of the relations class (df-rels 38503) 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 38504	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5661	. 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 3459   ⊆ wss 3926   × cxp 5652  Rel wrel 5659   Rels crels 38201

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ss 3943  df-pw 4577  df-rel 5661  df-rels 38503

This theorem is referenced by:  elrelsrelim  38506  elrels5  38507  elrels6  38508  cnvelrels  38513  cosselrels  38514  elrefrelsrel  38538  elcnvrefrelsrel  38554  elsymrelsrel  38575  eltrrelsrel  38599  eleqvrelsrel  38612  elfunsALTVfunALTV  38715  eldisjs5  38744  eldisjsdisj  38745