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Definition df-eqvrel 36250
 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 36249) is equivalent to satisfying the equivalence relation predicate, see eleqvrelsrel 36259. Alternate definitions are dfeqvrel2 36255 and dfeqvrel3 36256. (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 35900 . 2 wff EqvRel 𝑅
31wrefrel 35889 . . 3 wff RefRel 𝑅
41wsymrel 35895 . . 3 wff SymRel 𝑅
51wtrrel 35898 . . 3 wff TrRel 𝑅
63, 4, 5w3a 1085 . 2 wff ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)
72, 6wb 209 1 wff ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅))
 Colors of variables: wff setvar class This definition is referenced by:  dfeqvrel2  36255  dfeqvrel3  36256  eqvrelrefrel  36263  eqvrelsymrel  36264  eqvreltrrel  36265  eqvreleq  36267  eqvrelcoss  36282  refrelredund2  36301
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