Description: Define the class of symmetric relations. For sets, being an element of the class of symmetric relations is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel . Alternate definitions are dfsymrels2 , dfsymrels3 , dfsymrels4 and dfsymrels5 .
This definition is similar to the definitions of the classes of reflexive ( df-refrels ) and transitive ( df-trrels ) relations. (Contributed by Peter Mazsa, 7-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | df-symrels | ⊢ SymRels = ( Syms ∩ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | csymrels | ⊢ SymRels | |
1 | csyms | ⊢ Syms | |
2 | crels | ⊢ Rels | |
3 | 1 2 | cin | ⊢ ( Syms ∩ Rels ) |
4 | 0 3 | wceq | ⊢ SymRels = ( Syms ∩ Rels ) |