Theorem elrels2 38565

Description: The element of the relations class (df-rels 38564) and the relation predicate (df-rel 5629) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)

Assertion

Ref	Expression
elrels2	⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Proof of Theorem elrels2

Step	Hyp	Ref	Expression
1		df-rels 38564	. . 3 ⊢ Rels = 𝒫 (V × V)
2	1	eleq2i 2826	. 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3		elpwg 4555	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V)))
4	2, 3	bitrid 283	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  Vcvv 3438   ⊆ wss 3899  𝒫 cpw 4552   × cxp 5620   Rels crels 38324

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ss 3916  df-pw 4554  df-rels 38564

This theorem is referenced by:  elrelsrel  38566