Theorem elrelsrel 37357

Description: The element of the relations class (df-rels 37355) 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 37356	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5684	. 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 2107  Vcvv 3475   ⊆ wss 3949   × cxp 5675  Rel wrel 5682   Rels crels 37045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-rel 5684  df-rels 37355

This theorem is referenced by:  elrelsrelim  37358  elrels5  37359  elrels6  37360  cnvelrels  37365  cosselrels  37366  elrefrelsrel  37390  elcnvrefrelsrel  37406  elsymrelsrel  37427  eltrrelsrel  37451  eleqvrelsrel  37464  elfunsALTVfunALTV  37567  eldisjs5  37596  eldisjsdisj  37597