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Definition df-rcl 38484
Description: Reflexive closure of a relation. This is the smallest superset which has the reflexive property. (Contributed by RP, 5-Jun-2020.)
Assertion
Ref Expression
df-rcl r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
Distinct variable group:   𝑥,𝑧

Detailed syntax breakdown of Definition df-rcl
StepHypRef Expression
1 crcl 38483 . 2 class r*
2 vx . . 3 setvar 𝑥
3 cvv 3349 . . 3 class V
42cv 1629 . . . . . . 7 class 𝑥
5 vz . . . . . . . 8 setvar 𝑧
65cv 1629 . . . . . . 7 class 𝑧
74, 6wss 3721 . . . . . 6 wff 𝑥𝑧
8 cid 5156 . . . . . . . 8 class I
96cdm 5249 . . . . . . . . 9 class dom 𝑧
106crn 5250 . . . . . . . . 9 class ran 𝑧
119, 10cun 3719 . . . . . . . 8 class (dom 𝑧 ∪ ran 𝑧)
128, 11cres 5251 . . . . . . 7 class ( I ↾ (dom 𝑧 ∪ ran 𝑧))
1312, 6wss 3721 . . . . . 6 wff ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧
147, 13wa 382 . . . . 5 wff (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)
1514, 5cab 2756 . . . 4 class {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
1615cint 4609 . . 3 class {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
172, 3, 16cmpt 4861 . 2 class (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
181, 17wceq 1630 1 wff r* = (𝑥 ∈ V ↦ {𝑧 ∣ (𝑥𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
Colors of variables: wff setvar class
This definition is referenced by:  dfrcl2  38485
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