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Definition df-cnvrefrel 35925
Description: Define the converse reflexive relation predicate (read: 𝑅 is a converse reflexive relation), see also the comment of dfcnvrefrel3 35929. Alternate definitions are dfcnvrefrel2 35928 and dfcnvrefrel3 35929. (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
StepHypRef Expression
1 cR . . 3 class 𝑅
21wcnvrefrel 35622 . 2 wff CnvRefRel 𝑅
31cdm 5519 . . . . . 6 class dom 𝑅
41crn 5520 . . . . . 6 class ran 𝑅
53, 4cxp 5517 . . . . 5 class (dom 𝑅 × ran 𝑅)
61, 5cin 3880 . . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
7 cid 5424 . . . . 5 class I
87, 5cin 3880 . . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅))
96, 8wss 3881 . . 3 wff (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅))
101wrel 5524 . . 3 wff Rel 𝑅
119, 10wa 399 . 2 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
122, 11wb 209 1 wff ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Colors of variables: wff setvar class
This definition is referenced by:  dfcnvrefrel2  35928  dfcnvrefrel3  35929
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