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Theorem elrels2 35831
Description: The element of the relations class (df-rels 35830) and the relation predicate (df-rel 5549) are the same when 𝑅 is a set. (Contributed by Peter Mazsa, 14-Jun-2018.)
Assertion
Ref Expression
elrels2 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))

Proof of Theorem elrels2
StepHypRef Expression
1 df-rels 35830 . . 3 Rels = 𝒫 (V × V)
21eleq2i 2907 . 2 (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3 elpwg 4525 . 2 (𝑅𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V)))
42, 3syl5bb 286 1 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2115  Vcvv 3480  wss 3919  𝒫 cpw 4522   × cxp 5540   Rels crels 35560
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-pw 4524  df-rels 35830
This theorem is referenced by:  elrelsrel  35832
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