Theorem elrelsrel 38503

Description: The element of the relations class (df-rels 38501) 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 38502	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5621	. 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 2110  Vcvv 3434   ⊆ wss 3900   × cxp 5612  Rel wrel 5619   Rels crels 38196

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 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ss 3917  df-pw 4550  df-rel 5621  df-rels 38501

This theorem is referenced by:  elrelsrelim  38504  elrels5  38505  elrels6  38506  cnvelrels  38511  cosselrels  38512  elrefrelsrel  38536  elcnvrefrelsrel  38552  elsymrelsrel  38573  eltrrelsrel  38597  eleqvrelsrel  38610  elfunsALTVfunALTV  38714  eldisjs5  38743  eldisjsdisj  38744