Theorem dfrtrcl3 43723

Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15098. (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 15024	. 2 ⊢ t* = (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		relexp0g 15058	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3		nn0ex 12530	. . . . . . . . 9 ⊢ ℕ0 ∈ V
4		0nn0 12539	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
5		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → (𝑎↑𝑟𝑛) = (𝑡↑𝑟𝑛))
6	5	iuneq2d 5027	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛))
7		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑡↑𝑟𝑛) = (𝑡↑𝑟𝑘))
8	7	cbviunv 5045	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘)
9	6, 8	eqtrdi 2791	. . . . . . . . . . 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 43690	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
12	3, 4, 11	mp3an23 1452	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
13	2, 12	eqsstrrd 4035	. . . . . . 7 ⊢ (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
14		relexp1g 15062	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) = 𝑟)
15		1nn0 12540	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
16	10	ov2ssiunov2 43690	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
17	3, 15, 16	mp3an23 1452	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
18	14, 17	eqsstrrd 4035	. . . . . . 7 ⊢ (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
19		nn0uz 12918	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
20	10	iunrelexpuztr 43709	. . . . . . . 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 1452	. . . . . . 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 6920	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V
23		sseq2 4022	. . . . . . . . . . 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 4022	. . . . . . . . . . 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 5878	. . . . . . . . . . . 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 4029	. . . . . . . . . . 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 1435	. . . . . . . . . 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 1925	. . . . . . . 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 3672	. . . . . . . 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 1341	. . . . . 6 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
34		intss1 4968	. . . . . 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 3482	. . . . . . . . 9 ⊢ 𝑠 ∈ V
37		sseq2 4022	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38		sseq2 4022	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ 𝑠))
39		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
40	39, 39	coeq12d 5878	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
41	40, 39	sseq12d 4029	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
42	37, 38, 41	3anbi123d 1435	. . . . . . . . 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 3681	. . . . . . . 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 2735	. . . . . . . . . 10 ⊢ ℕ0 = ℕ0
45	10	iunrelexpmin2 43702	. . . . . . . . . 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 691	. . . . . . . . 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 2187	. . . . . . . 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 242	. . . . . . 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 3143	. . . . . 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 4969	. . . . . 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 234	. . . . 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 4013	. . . 4 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
53		oveq1 7438	. . . . . 6 ⊢ (𝑎 = 𝑟 → (𝑎↑𝑟𝑛) = (𝑟↑𝑟𝑛))
54	53	iuneq2d 5027	. . . . 5 ⊢ (𝑎 = 𝑟 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
55		eqid 2735	. . . . 5 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))
56		ovex 7464	. . . . . 6 ⊢ (𝑟↑𝑟𝑛) ∈ V
57	3, 56	iunex 7992	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) ∈ V
58	54, 55, 57	fvmpt 7016	. . . 4 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
59	52, 58	eqtrd 2775	. . 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 2763	1 ⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086  ∀wal 1535   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  Vcvv 3478   ∪ cun 3961   ⊆ wss 3963  ∩ cint 4951  ∪ ciun 4996   ↦ cmpt 5231   I cid 5582  dom cdm 5689  ran crn 5690   ↾ cres 5691   ∘ ccom 5693  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  ℕ₀cn0 12524  ℤ_≥cuz 12876  t*crtcl 15022  ↑_𝑟crelexp 15055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-seq 14040  df-rtrcl 15024  df-relexp 15056

This theorem is referenced by:  brfvrtrcld  43724  fvrtrcllb0d  43725  fvrtrcllb0da  43726  fvrtrcllb1d  43727  dfrtrcl4  43728  corcltrcl  43729  cotrclrcl  43732