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 ) |
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 ) |