Theorem elrelsrel 39483

Description: The element of the relations class (df-rels 39481) 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 39482	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5658	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 292	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   × cxp 5649  Rel wrel 5656   Rels crels 39226

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3897  df-pw 4558  df-rel 5658  df-rels 39481

This theorem is used by:  elrelsrelim  39484  elrels5  39485  elrels6  39486  cosselrels  39616  cnvelrels  39617  elrefrelsrel  39641  elcnvrefrelsrel  39657  elsymrelsrel  39682  eltrrelsrel  39706  eleqvrelsrel  39719  elfunsALTVfunALTV  39823  eldisjs5  39864  eldisjsdisj  39865