Theorem elrels2 38692

Description: The element of the relations class (df-rels 38691) and the relation predicate (df-rel 5639) 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 38691	. . 3 ⊢ Rels = 𝒫 (V × V)
2	1	eleq2i 2829	. 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3		elpwg 4559	. 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 2114  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   × cxp 5630   Rels crels 38436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-pw 4558  df-rels 38691

This theorem is referenced by:  elrelsrel  38693