Metamath Proof Explorer


Definition df-cnvrefrel

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

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

Detailed syntax breakdown

Step Hyp Ref Expression
0 cR 𝑅
1 0 wcnvrefrel CnvRefRel 𝑅
2 0 cdm dom 𝑅
3 0 crn ran 𝑅
4 2 3 cxp ( dom 𝑅 × ran 𝑅 )
5 0 4 cin ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) )
6 cid I
7 6 4 cin ( I ∩ ( dom 𝑅 × ran 𝑅 ) )
8 5 7 wss ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) )
9 0 wrel Rel 𝑅
10 8 9 wa ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 )
11 1 10 wb ( CnvRefRel 𝑅 ↔ ( ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( I ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )