Theorem elrelsrel 37999

Description: The element of the relations class (df-rels 37997) 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 37998	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5689	. 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 2098  Vcvv 3473   ⊆ wss 3949   × cxp 5680  Rel wrel 5687   Rels crels 37691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-pw 4608  df-rel 5689  df-rels 37997

This theorem is referenced by:  elrelsrelim  38000  elrels5  38001  elrels6  38002  cnvelrels  38007  cosselrels  38008  elrefrelsrel  38032  elcnvrefrelsrel  38048  elsymrelsrel  38069  eltrrelsrel  38093  eleqvrelsrel  38106  elfunsALTVfunALTV  38209  eldisjs5  38238  eldisjsdisj  38239