Theorem elrelsrel 38469

Description: The element of the relations class (df-rels 38467) 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 38468	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5696	. 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 2106  Vcvv 3478   ⊆ wss 3963   × cxp 5687  Rel wrel 5694   Rels crels 38164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ss 3980  df-pw 4607  df-rel 5696  df-rels 38467

This theorem is referenced by:  elrelsrelim  38470  elrels5  38471  elrels6  38472  cnvelrels  38477  cosselrels  38478  elrefrelsrel  38502  elcnvrefrelsrel  38518  elsymrelsrel  38539  eltrrelsrel  38563  eleqvrelsrel  38576  elfunsALTVfunALTV  38679  eldisjs5  38708  eldisjsdisj  38709