Theorem elrels2 38477

Description: The element of the relations class (df-rels 38476) and the relation predicate (df-rel 5645) 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 38476	. . 3 ⊢ Rels = 𝒫 (V × V)
2	1	eleq2i 2820	. 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3		elpwg 4566	. 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 2109  Vcvv 3447   ⊆ wss 3914  𝒫 cpw 4563   × cxp 5636   Rels crels 38171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ss 3931  df-pw 4565  df-rels 38476

This theorem is referenced by:  elrelsrel  38478