Theorem elrelsrel 38530

Description: The element of the relations class (df-rels 38528) 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 38529	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5623	. 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 2111  Vcvv 3436   ⊆ wss 3902   × cxp 5614  Rel wrel 5621   Rels crels 38223

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ss 3919  df-pw 4552  df-rel 5623  df-rels 38528

This theorem is referenced by:  elrelsrelim  38531  elrels5  38532  elrels6  38533  cnvelrels  38538  cosselrels  38539  elrefrelsrel  38563  elcnvrefrelsrel  38579  elsymrelsrel  38600  eltrrelsrel  38624  eleqvrelsrel  38637  elfunsALTVfunALTV  38741  eldisjs5  38770  eldisjsdisj  38771