Metamath Proof Explorer


Theorem eqvrelim

Description: Equivalence relation implies that the domain and the range are equal. (Contributed by Peter Mazsa, 29-Dec-2021)

Ref Expression
Assertion eqvrelim ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅 )

Proof

Step Hyp Ref Expression
1 eqvrelsymrel ( EqvRel 𝑅 → SymRel 𝑅 )
2 symrelim ( SymRel 𝑅 → dom 𝑅 = ran 𝑅 )
3 1 2 syl ( EqvRel 𝑅 → dom 𝑅 = ran 𝑅 )