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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-symrels | Structured version Visualization version GIF version | ||
| 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 38558. Alternate definitions are
dfsymrels2 38546, dfsymrels3 38547, dfsymrels4 38548 and dfsymrels5 38549.
This definition is similar to the definitions of the classes of reflexive (df-refrels 38512) and transitive (df-trrels 38574) relations. (Contributed by Peter Mazsa, 7-Jul-2019.) |
| Ref | Expression |
|---|---|
| df-symrels | ⊢ SymRels = ( Syms ∩ Rels ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csymrels 38193 | . 2 class SymRels | |
| 2 | csyms 38192 | . . 3 class Syms | |
| 3 | crels 38184 | . . 3 class Rels | |
| 4 | 2, 3 | cin 3950 | . 2 class ( Syms ∩ Rels ) |
| 5 | 1, 4 | wceq 1540 | 1 wff SymRels = ( Syms ∩ Rels ) |
| Colors of variables: wff setvar class |
| This definition is referenced by: dfsymrels2 38546 |
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