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Theorem elrels2 37990
Description: The element of the relations class (df-rels 37989) and the relation predicate (df-rel 5689) 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 37989 . . 3 Rels = 𝒫 (V × V)
21eleq2i 2821 . 2 (𝑅 ∈ Rels ↔ 𝑅 ∈ 𝒫 (V × V))
3 elpwg 4609 . 2 (𝑅𝑉 → (𝑅 ∈ 𝒫 (V × V) ↔ 𝑅 ⊆ (V × V)))
42, 3bitrid 282 1 (𝑅𝑉 → (𝑅 ∈ Rels ↔ 𝑅 ⊆ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  Vcvv 3473  wss 3949  𝒫 cpw 4606   × cxp 5680   Rels crels 37683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-pw 4608  df-rels 37989
This theorem is referenced by:  elrelsrel  37991
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