Theorem elrelsrel 37870

Description: The element of the relations class (df-rels 37868) 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 37869	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5676	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 289	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943   × cxp 5667  Rel wrel 5674   Rels crels 37558

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-rel 5676  df-rels 37868

This theorem is referenced by:  elrelsrelim  37871  elrels5  37872  elrels6  37873  cnvelrels  37878  cosselrels  37879  elrefrelsrel  37903  elcnvrefrelsrel  37919  elsymrelsrel  37940  eltrrelsrel  37964  eleqvrelsrel  37977  elfunsALTVfunALTV  38080  eldisjs5  38109  eldisjsdisj  38110