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Definition df-eqvrel 37450
Description: Define the equivalence relation predicate. (Read: 𝑅 is an equivalence relation.) For sets, being an element of the class of equivalence relations (df-eqvrels 37449) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 37459. Alternate definitions are dfeqvrel2 37455 and dfeqvrel3 37456. (Contributed by Peter Mazsa, 17-Apr-2019.)
Assertion
Ref Expression
df-eqvrel ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))

Detailed syntax breakdown of Definition df-eqvrel
StepHypRef Expression
1 cR . . 3 class 𝑅
21weqvrel 37055 . 2 wff EqvRel 𝑅
31wrefrel 37044 . . 3 wff RefRel 𝑅
41wsymrel 37050 . . 3 wff SymRel 𝑅
51wtrrel 37053 . . 3 wff TrRel 𝑅
63, 4, 5w3a 1087 . 2 wff ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)
72, 6wb 205 1 wff ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
Colors of variables: wff setvar class
This definition is referenced by:  dfeqvrel2  37455  dfeqvrel3  37456  eqvrelrefrel  37463  eqvrelsymrel  37464  eqvreltrrel  37465  eqvreleq  37467  eqvrelcoss  37482  refrelredund2  37501
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