Theorem elrelsrel 35729

Description: The element of the relations class (df-rels 35727) 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 35728	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
2		df-rel 5564	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	syl6bbr 291	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938   × cxp 5555  Rel wrel 5562   Rels crels 35457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-in 3945  df-ss 3954  df-pw 4543  df-rel 5564  df-rels 35727

This theorem is referenced by:  elrelsrelim  35730  elrels5  35731  elrels6  35732  cnvelrels  35737  cosselrels  35738  elrefrelsrel  35761  elcnvrefrelsrel  35774  elsymrelsrel  35795  eltrrelsrel  35819  eleqvrelsrel  35831  elfunsALTVfunALTV  35932  eldisjs5  35961  eldisjsdisj  35962