Theorem elrelsrel 36532

Description: The element of the relations class (df-rels 36530) 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 36531	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5587	. 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 2108  Vcvv 3422   ⊆ wss 3883   × cxp 5578  Rel wrel 5585   Rels crels 36262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-rel 5587  df-rels 36530

This theorem is referenced by:  elrelsrelim  36533  elrels5  36534  elrels6  36535  cnvelrels  36540  cosselrels  36541  elrefrelsrel  36564  elcnvrefrelsrel  36577  elsymrelsrel  36598  eltrrelsrel  36622  eleqvrelsrel  36634  elfunsALTVfunALTV  36735  eldisjs5  36764  eldisjsdisj  36765