Theorem elrelsrel 39119

Description: The element of the relations class (df-rels 39117) 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 39118	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5668	. 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 2143  Vcvv 3455   ⊆ wss 3905   × cxp 5659  Rel wrel 5666   Rels crels 38862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ss 3922  df-pw 4564  df-rel 5668  df-rels 39117

This theorem is used by:  elrelsrelim  39120  elrels5  39121  elrels6  39122  cosselrels  39252  cnvelrels  39253  elrefrelsrel  39277  elcnvrefrelsrel  39293  elsymrelsrel  39318  eltrrelsrel  39342  eleqvrelsrel  39355  elfunsALTVfunALTV  39459  eldisjs5  39500  eldisjsdisj  39501