Theorem elrels2 38444

Description: The element of the relations class (df-rels 38443) and the relation predicate (df-rel 5707) 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 38443	. . 3 ⊢ Rels = 𝒫 (V × V)
2	1	eleq2i 2836	. 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3		elpwg 4625	. 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 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622   × cxp 5698   Rels crels 38139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ss 3993  df-pw 4624  df-rels 38443

This theorem is referenced by:  elrelsrel  38445