Theorem elrelsrel 38195

Description: The element of the relations class (df-rels 38193) 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 38194	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5679	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 288	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2099  Vcvv 3462   ⊆ wss 3946   × cxp 5670  Rel wrel 5677   Rels crels 37888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ss 3963  df-pw 4599  df-rel 5679  df-rels 38193

This theorem is referenced by:  elrelsrelim  38196  elrels5  38197  elrels6  38198  cnvelrels  38203  cosselrels  38204  elrefrelsrel  38228  elcnvrefrelsrel  38244  elsymrelsrel  38265  eltrrelsrel  38289  eleqvrelsrel  38302  elfunsALTVfunALTV  38405  eldisjs5  38434  eldisjsdisj  38435