Theorem dfrtrcl3 42474

Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15008. (Contributed by RP, 5-Jun-2020.)

Assertion

Ref	Expression
dfrtrcl3	⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Distinct variable group:   𝑛,𝑟

Proof of Theorem dfrtrcl3

Dummy variables 𝑘 𝑎 𝑡 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rtrcl 14934	. 2 ⊢ t* = (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		relexp0g 14968	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3		nn0ex 12477	. . . . . . . . 9 ⊢ ℕ0 ∈ V
4		0nn0 12486	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
5		oveq1 7415	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → (𝑎↑𝑟𝑛) = (𝑡↑𝑟𝑛))
6	5	iuneq2d 5026	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛))
7		oveq2 7416	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑡↑𝑟𝑛) = (𝑡↑𝑟𝑘))
8	7	cbviunv 5043	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘)
9	6, 8	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
10	9	cbvmptv 5261	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑡 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
11	10	ov2ssiunov2 42441	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
12	3, 4, 11	mp3an23 1453	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
13	2, 12	eqsstrrd 4021	. . . . . . 7 ⊢ (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
14		relexp1g 14972	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) = 𝑟)
15		1nn0 12487	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
16	10	ov2ssiunov2 42441	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
17	3, 15, 16	mp3an23 1453	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
18	14, 17	eqsstrrd 4021	. . . . . . 7 ⊢ (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
19		nn0uz 12863	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
20	10	iunrelexpuztr 42460	. . . . . . . 8 ⊢ ((𝑟 ∈ V ∧ ℕ0 = (ℤ≥‘0) ∧ 0 ∈ ℕ0) → (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
21	19, 4, 20	mp3an23 1453	. . . . . . 7 ⊢ (𝑟 ∈ V → (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
22		fvex 6904	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V
23		sseq2 4008	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))
24		sseq2 4008	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))
25		id 22	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → 𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
26	25, 25	coeq12d 5864	. . . . . . . . . . . 12 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → (𝑧 ∘ 𝑧) = (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))
27	26, 25	sseq12d 4015	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))
28	23, 24, 27	3anbi123d 1436	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))))
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑟 ∈ V → (𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))))
30	29	alrimiv 1930	. . . . . . . 8 ⊢ (𝑟 ∈ V → ∀𝑧(𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)))))
31		elabgt 3662	. . . . . . . 8 ⊢ ((((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V ∧ ∀𝑧(𝑧 = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))))) → (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))))
32	22, 30, 31	sylancr 587	. . . . . . 7 ⊢ (𝑟 ∈ V → (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∧ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∘ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))))
33	13, 18, 21, 32	mpbir3and 1342	. . . . . 6 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
34		intss1 4967	. . . . . 6 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
35	33, 34	syl 17	. . . . 5 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
36		vex 3478	. . . . . . . . 9 ⊢ 𝑠 ∈ V
37		sseq2 4008	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38		sseq2 4008	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ 𝑠))
39		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
40	39, 39	coeq12d 5864	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
41	40, 39	sseq12d 4015	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
42	37, 38, 41	3anbi123d 1436	. . . . . . . . 9 ⊢ (𝑧 = 𝑠 → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠 ∧ 𝑟 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)))
43	36, 42	elab 3668	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠 ∧ 𝑟 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠))
44		eqid 2732	. . . . . . . . . 10 ⊢ ℕ0 = ℕ0
45	10	iunrelexpmin2 42453	. . . . . . . . . 10 ⊢ ((𝑟 ∈ V ∧ ℕ0 = ℕ0) → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠 ∧ 𝑟 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠))
46	44, 45	mpan2 689	. . . . . . . . 9 ⊢ (𝑟 ∈ V → ∀𝑠((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠 ∧ 𝑟 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠))
47	46	19.21bi 2182	. . . . . . . 8 ⊢ (𝑟 ∈ V → ((( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠 ∧ 𝑟 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠))
48	43, 47	biimtrid 241	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠))
49	48	ralrimiv 3145	. . . . . 6 ⊢ (𝑟 ∈ V → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠)
50		ssint 4968	. . . . . 6 ⊢ (((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ 𝑠)
51	49, 50	sylibr 233	. . . . 5 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
52	35, 51	eqssd 3999	. . . 4 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
53		oveq1 7415	. . . . . 6 ⊢ (𝑎 = 𝑟 → (𝑎↑𝑟𝑛) = (𝑟↑𝑟𝑛))
54	53	iuneq2d 5026	. . . . 5 ⊢ (𝑎 = 𝑟 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
55		eqid 2732	. . . . 5 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))
56		ovex 7441	. . . . . 6 ⊢ (𝑟↑𝑟𝑛) ∈ V
57	3, 56	iunex 7954	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) ∈ V
58	54, 55, 57	fvmpt 6998	. . . 4 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
59	52, 58	eqtrd 2772	. . 3 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
60	59	mpteq2ia 5251	. 2 ⊢ (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
61	1, 60	eqtri 2760	1 ⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1087  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  Vcvv 3474   ∪ cun 3946   ⊆ wss 3948  ∩ cint 4950  ∪ ciun 4997   ↦ cmpt 5231   I cid 5573  dom cdm 5676  ran crn 5677   ↾ cres 5678   ∘ ccom 5680  ‘cfv 6543  (class class class)co 7408  0cc0 11109  1c1 11110  ℕ₀cn0 12471  ℤ_≥cuz 12821  t*crtcl 14932  ↑_𝑟crelexp 14965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-seq 13966  df-rtrcl 14934  df-relexp 14966

This theorem is referenced by:  brfvrtrcld  42475  fvrtrcllb0d  42476  fvrtrcllb0da  42477  fvrtrcllb1d  42478  dfrtrcl4  42479  corcltrcl  42480  cotrclrcl  42483