Metamath Proof Explorer


Definition df-refrel

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

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

Detailed syntax breakdown

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