Description: Define the class of converse reflexive relations. This is practically dfcnvrefrels2 (which uses the traditional subclass relation C_ ) : we use converse subset relation ( brcnvssr ) here to ensure the comparability to the definitions of the classes of all reflexive ( df-ref ), symmetric ( df-syms ) and transitive ( df-trs ) sets.
We use this concept to define functions ( df-funsALTV , df-funALTV ) and disjoints ( df-disjs , df-disjALTV ).
For sets, being an element of the class of converse reflexive relations is equivalent to satisfying the converse reflexive relation predicate, see elcnvrefrelsrel . Alternate definitions are dfcnvrefrels2 and dfcnvrefrels3 . (Contributed by Peter Mazsa, 7-Jul-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | df-cnvrefrels | ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | ccnvrefrels | ⊢ CnvRefRels | |
1 | ccnvrefs | ⊢ CnvRefs | |
2 | crels | ⊢ Rels | |
3 | 1 2 | cin | ⊢ ( CnvRefs ∩ Rels ) |
4 | 0 3 | wceq | ⊢ CnvRefRels = ( CnvRefs ∩ Rels ) |