Definition df-rtrcl 13575

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)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})

Distinct variable group:   𝑥,𝑧

Detailed syntax breakdown of Definition df-rtrcl

Step	Hyp	Ref	Expression
1		crtcl 13573	. 2 class t*
2		vx	. . 3 setvar 𝑥
3		cvv 3173	. . 3 class V
4		cid 4948	. . . . . . . 8 class I
5	2	cv 1474	. . . . . . . . . 10 class 𝑥
6	5	cdm 5038	. . . . . . . . 9 class dom 𝑥
7	5	crn 5039	. . . . . . . . 9 class ran 𝑥
8	6, 7	cun 3538	. . . . . . . 8 class (dom 𝑥 ∪ ran 𝑥)
9	4, 8	cres 5040	. . . . . . 7 class ( I ↾ (dom 𝑥 ∪ ran 𝑥))
10		vz	. . . . . . . 8 setvar 𝑧
11	10	cv 1474	. . . . . . 7 class 𝑧
12	9, 11	wss 3540	. . . . . 6 wff ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧
13	5, 11	wss 3540	. . . . . 6 wff 𝑥 ⊆ 𝑧
14	11, 11	ccom 5042	. . . . . . 7 class (𝑧 ∘ 𝑧)
15	14, 11	wss 3540	. . . . . 6 wff (𝑧 ∘ 𝑧) ⊆ 𝑧
16	12, 13, 15	w3a 1031	. . . . 5 wff (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)
17	16, 10	cab 2596	. . . 4 class {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
18	17	cint 4410	. . 3 class ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}
19	2, 3, 18	cmpt 4643	. 2 class (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
20	1, 19	wceq 1475	1 wff t* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})

This definition is referenced by:  dfrtrcl2  13650  dfrtrcl5  36955  dfrtrcl3  37044