Theorem elrelsrel 39124

Description: The element of the relations class (df-rels 39122) 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 39123	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5670	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4di 292	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906   × cxp 5661  Rel wrel 5668   Rels crels 38867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ss 3923  df-pw 4566  df-rel 5670  df-rels 39122

This theorem is used by:  elrelsrelim  39125  elrels5  39126  elrels6  39127  cosselrels  39257  cnvelrels  39258  elrefrelsrel  39282  elcnvrefrelsrel  39298  elsymrelsrel  39323  eltrrelsrel  39347  eleqvrelsrel  39360  elfunsALTVfunALTV  39464  eldisjs5  39505  eldisjsdisj  39506