Theorem elrefsymrels3

Description: Elements of the class of reflexive relations which are elements of the class of symmetric relations as well (like the elements of the class of equivalence relations dfeqvrels3 ) can use the A. x e. dom R x R x version for their reflexive part, not just the A. x e. dom R A. y e. ran R ( x = y -> x R y ) version of dfrefrels3 , cf. the comment of dfrefrel3 . (Contributed by Peter Mazsa, 22-Jul-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	elrefsymrels3	|- ( R e. ( RefRels i^i SymRels ) <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) /\ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		elrefsymrels2	|- ( R e. ( RefRels i^i SymRels ) <-> ( ( ( _I |` dom R ) C_ R /\ `' R C_ R ) /\ R e. Rels ) )
2		idrefALT	|- ( ( _I |` dom R ) C_ R <-> A. x e. dom R x R x )
3		cnvsym	|- ( `' R C_ R <-> A. x A. y ( x R y -> y R x ) )
4	2 3	anbi12i	|- ( ( ( _I |` dom R ) C_ R /\ `' R C_ R ) <-> ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) )
5	4	anbi1i	|- ( ( ( ( _I |` dom R ) C_ R /\ `' R C_ R ) /\ R e. Rels ) <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) /\ R e. Rels ) )
6	1 5	bitri	|- ( R e. ( RefRels i^i SymRels ) <-> ( ( A. x e. dom R x R x /\ A. x A. y ( x R y -> y R x ) ) /\ R e. Rels ) )