Definition df-refrel 39103

Description: Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 39107. Alternate definitions are dfrefrel2 39106 and dfrefrel3 39107. For sets, being an element of the class of reflexive relations (df-refrels 39102) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 39111. (Contributed by Peter Mazsa, 16-Jul-2021.)

Assertion

Ref	Expression
df-refrel	⊢ ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-refrel

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	wrefrel 38700	. 2 wff RefRel 𝑅
3		cid 5546	. . . . 5 class I
4	1	cdm 5652	. . . . . 6 class dom 𝑅
5	1	crn 5653	. . . . . 6 class ran 𝑅
6	4, 5	cxp 5650	. . . . 5 class (dom 𝑅 × ran 𝑅)
7	3, 6	cin 3906	. . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅))
8	1, 6	cin 3906	. . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
9	7, 8	wss 3907	. . 3 wff ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅))
10	1	wrel 5657	. . 3 wff Rel 𝑅
11	9, 10	wa 400	. 2 wff (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
12	2, 11	wb 209	1 wff ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

This definition is referenced by:  dfrefrel2  39106  refrelid  39113