Definition df-cnvrefrel 38508

Description: Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 38512. Alternate definitions are dfcnvrefrel2 38511 and dfcnvrefrel3 38512. (Contributed by Peter Mazsa, 16-Jul-2021.)

Assertion

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

Detailed syntax breakdown of Definition df-cnvrefrel

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	wcnvrefrel 38170	. 2 wff CnvRefRel 𝑅
3	1	cdm 5688	. . . . . 6 class dom 𝑅
4	1	crn 5689	. . . . . 6 class ran 𝑅
5	3, 4	cxp 5686	. . . . 5 class (dom 𝑅 × ran 𝑅)
6	1, 5	cin 3961	. . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
7		cid 5581	. . . . 5 class I
8	7, 5	cin 3961	. . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅))
9	6, 8	wss 3962	. . 3 wff (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))
10	1	wrel 5693	. . 3 wff Rel 𝑅
11	9, 10	wa 395	. 2 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
12	2, 11	wb 206	1 wff ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

This definition is referenced by:  dfcnvrefrel2  38511  dfcnvrefrel3  38512  dfcnvrefrel4  38513