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Definition df-rtrcl 13677
Description: Reflexive-transitive closure of a relation. This is the smallest superset which is reflexive property over all elements of its domain and range and has the transitive property. (Contributed by FL, 27-Jun-2011.)
Assertion
Ref Expression
df-rtrcl t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
Distinct variable group:   𝑥,𝑧

Detailed syntax breakdown of Definition df-rtrcl
StepHypRef Expression
1 crtcl 13675 . 2 class t*
2 vx . . 3 setvar 𝑥
3 cvv 3190 . . 3 class V
4 cid 4994 . . . . . . . 8 class I
52cv 1479 . . . . . . . . . 10 class 𝑥
65cdm 5084 . . . . . . . . 9 class dom 𝑥
75crn 5085 . . . . . . . . 9 class ran 𝑥
86, 7cun 3558 . . . . . . . 8 class (dom 𝑥 ∪ ran 𝑥)
94, 8cres 5086 . . . . . . 7 class ( I ↾ (dom 𝑥 ∪ ran 𝑥))
10 vz . . . . . . . 8 setvar 𝑧
1110cv 1479 . . . . . . 7 class 𝑧
129, 11wss 3560 . . . . . 6 wff ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧
135, 11wss 3560 . . . . . 6 wff 𝑥𝑧
1411, 11ccom 5088 . . . . . . 7 class (𝑧𝑧)
1514, 11wss 3560 . . . . . 6 wff (𝑧𝑧) ⊆ 𝑧
1612, 13, 15w3a 1036 . . . . 5 wff (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)
1716, 10cab 2607 . . . 4 class {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
1817cint 4447 . . 3 class {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)}
192, 3, 18cmpt 4683 . 2 class (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
201, 19wceq 1480 1 wff t* = (𝑥 ∈ V ↦ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧𝑥𝑧 ∧ (𝑧𝑧) ⊆ 𝑧)})
Colors of variables: wff setvar class
This definition is referenced by:  dfrtrcl2  13752  dfrtrcl5  37456  dfrtrcl3  37545
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