Theorem elrels2 39110

Description: The element of the relations class (df-rels 39109) and the relation predicate (df-rel 5668) 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 39109	. . 3 ⊢ Rels = 𝒫 (V × V)
2	1	eleq2i 2855	. 2 ⊢ (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3		elpwg 4565	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V)))
4	2, 3	bitrid 286	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562   × cxp 5659   Rels crels 38854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ss 3922  df-pw 4564  df-rels 39109

This theorem is referenced by:  elrelsrel  39111