Definition df-refrel

Description: Define the reflexive relation predicate. (Read: R is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 . Alternate definitions are dfrefrel2 and dfrefrel3 . For sets, being an element of the class of reflexive relations ( df-refrels ) is equivalent to satisfying the reflexive relation predicate, that is ( R e. RefRels <-> RefRel R ) when R is a set, see elrefrelsrel . (Contributed by Peter Mazsa, 16-Jul-2021)

	Ref	Expression
Assertion	df-refrel	|- ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cR	|- R
1	0	wrefrel	|- RefRel R
2		cid	|- _I
3	0	cdm	|- dom R
4	0	crn	|- ran R
5	3 4	cxp	|- ( dom R X. ran R )
6	2 5	cin	|- ( _I i^i ( dom R X. ran R ) )
7	0 5	cin	|- ( R i^i ( dom R X. ran R ) )
8	6 7	wss	|- ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) )
9	0	wrel	|- Rel R
10	8 9	wa	|- ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R )
11	1 10	wb	|- ( RefRel R <-> ( ( _I i^i ( dom R X. ran R ) ) C_ ( R i^i ( dom R X. ran R ) ) /\ Rel R ) )