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Theorem elrels2 36994
Description: The element of the relations class (df-rels 36993) and the relation predicate (df-rel 5641) 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 36993 . . 3 Rels = 𝒫 (V × V)
21eleq2i 2826 . 2 (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3 elpwg 4564 . 2 (𝑅𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V)))
42, 3bitrid 283 1 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2107  Vcvv 3444  wss 3911  𝒫 cpw 4561   × cxp 5632   Rels crels 36682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563  df-rels 36993
This theorem is referenced by:  elrelsrel  36995
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