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Definition df-refrel 35812
Description: Define the reflexive relation predicate. (Read: 𝑅 is a reflexive relation.) This is a surprising definition, see the comment of dfrefrel3 35816. Alternate definitions are dfrefrel2 35815 and dfrefrel3 35816. For sets, being an element of the class of reflexive relations (df-refrels 35811) is equivalent to satisfying the reflexive relation predicate, that is (𝑅 ∈ RefRels ↔ RefRel 𝑅) when 𝑅 is a set, see elrefrelsrel 35819. (Contributed by Peter Mazsa, 16-Jul-2021.)
Assertion
Ref Expression
df-refrel ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-refrel
StepHypRef Expression
1 cR . . 3 class 𝑅
21wrefrel 35519 . 2 wff RefRel 𝑅
3 cid 5440 . . . . 5 class I
41cdm 5536 . . . . . 6 class dom 𝑅
51crn 5537 . . . . . 6 class ran 𝑅
64, 5cxp 5534 . . . . 5 class (dom 𝑅 × ran 𝑅)
73, 6cin 3917 . . . 4 class ( I ∩ (dom 𝑅 × ran 𝑅))
81, 6cin 3917 . . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
97, 8wss 3918 . . 3 wff ( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅))
101wrel 5541 . . 3 wff Rel 𝑅
119, 10wa 399 . 2 wff (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
122, 11wb 209 1 wff ( RefRel 𝑅 ↔ (( I ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Colors of variables: wff setvar class
This definition is referenced by:  dfrefrel2  35815  refrelid  35821
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