Definition df-cnvrefrel 39540

Description: Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 39544. Alternate definitions are dfcnvrefrel2 39543 and dfcnvrefrel3 39544. (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 39125	. 2 wff CnvRefRel 𝑅
3	1	cdm 5650	. . . . . 6 class dom 𝑅
4	1	crn 5651	. . . . . 6 class ran 𝑅
5	3, 4	cxp 5648	. . . . 5 class (dom 𝑅 × ran 𝑅)
6	1, 5	cin 4089	. . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
7		cid 5544	. . . . 5 class I
8	7, 5	cin 4089	. . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅))
9	6, 8	wss 3895	. . 3 wff (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))
10	1	wrel 5655	. . 3 wff Rel 𝑅
11	9, 10	wa 401	. 2 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
12	2, 11	wb 209	1 wff ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

This definition is used by:  dfcnvrefrel2  39543  dfcnvrefrel3  39544  dfcnvrefrel4  39545