Theorem dfrtrcl3 41230

Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14701. (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 14627	. 2 ⊢ t* = (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		relexp0g 14661	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3		nn0ex 12169	. . . . . . . . 9 ⊢ ℕ0 ∈ V
4		0nn0 12178	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
5		oveq1 7262	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → (𝑎↑𝑟𝑛) = (𝑡↑𝑟𝑛))
6	5	iuneq2d 4950	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛))
7		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑡↑𝑟𝑛) = (𝑡↑𝑟𝑘))
8	7	cbviunv 4966	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘)
9	6, 8	eqtrdi 2795	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
10	9	cbvmptv 5183	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑡 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
11	10	ov2ssiunov2 41197	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
12	3, 4, 11	mp3an23 1451	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
13	2, 12	eqsstrrd 3956	. . . . . . 7 ⊢ (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
14		relexp1g 14665	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) = 𝑟)
15		1nn0 12179	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
16	10	ov2ssiunov2 41197	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
17	3, 15, 16	mp3an23 1451	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
18	14, 17	eqsstrrd 3956	. . . . . . 7 ⊢ (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
19		nn0uz 12549	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
20	10	iunrelexpuztr 41216	. . . . . . . 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 1451	. . . . . . 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 6769	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V
23		sseq2 3943	. . . . . . . . . . 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 3943	. . . . . . . . . . 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 5762	. . . . . . . . . . . 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 3950	. . . . . . . . . . 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 1434	. . . . . . . . . 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 1931	. . . . . . . 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 3596	. . . . . . . 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 586	. . . . . . 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 1340	. . . . . 6 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
34		intss1 4891	. . . . . 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 3426	. . . . . . . . 9 ⊢ 𝑠 ∈ V
37		sseq2 3943	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38		sseq2 3943	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ 𝑠))
39		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
40	39, 39	coeq12d 5762	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
41	40, 39	sseq12d 3950	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
42	37, 38, 41	3anbi123d 1434	. . . . . . . . 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 3602	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ ℕ0 = ℕ0
45	10	iunrelexpmin2 41209	. . . . . . . . . 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 687	. . . . . . . . 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 2184	. . . . . . . 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	syl5bi 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 3106	. . . . . 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 4892	. . . . . 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 3934	. . . 4 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
53		oveq1 7262	. . . . . 6 ⊢ (𝑎 = 𝑟 → (𝑎↑𝑟𝑛) = (𝑟↑𝑟𝑛))
54	53	iuneq2d 4950	. . . . 5 ⊢ (𝑎 = 𝑟 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
55		eqid 2738	. . . . 5 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))
56		ovex 7288	. . . . . 6 ⊢ (𝑟↑𝑟𝑛) ∈ V
57	3, 56	iunex 7784	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) ∈ V
58	54, 55, 57	fvmpt 6857	. . . 4 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
59	52, 58	eqtrd 2778	. . 3 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
60	59	mpteq2ia 5173	. 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 2766	1 ⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ∪ cun 3881   ⊆ wss 3883  ∩ cint 4876  ∪ ciun 4921   ↦ cmpt 5153   I cid 5479  dom cdm 5580  ran crn 5581   ↾ cres 5582   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803  ℕ₀cn0 12163  ℤ_≥cuz 12511  t*crtcl 14625  ↑_𝑟crelexp 14658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-n0 12164  df-z 12250  df-uz 12512  df-seq 13650  df-rtrcl 14627  df-relexp 14659

This theorem is referenced by:  brfvrtrcld  41231  fvrtrcllb0d  41232  fvrtrcllb0da  41233  fvrtrcllb1d  41234  dfrtrcl4  41235  corcltrcl  41236  cotrclrcl  41239