Theorem dfrtrcl3 44181

Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 15018. (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 14944	. 2 ⊢ t* = (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		relexp0g 14978	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3		nn0ex 12437	. . . . . . . . 9 ⊢ ℕ0 ∈ V
4		0nn0 12446	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
5		oveq1 7368	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → (𝑎↑𝑟𝑛) = (𝑡↑𝑟𝑛))
6	5	iuneq2d 4965	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛))
7		oveq2 7369	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑡↑𝑟𝑛) = (𝑡↑𝑟𝑘))
8	7	cbviunv 4982	. . . . . . . . . . . 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 5190	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑡 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
11	10	ov2ssiunov2 44148	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
12	3, 4, 11	mp3an23 1456	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
13	2, 12	eqsstrrd 3958	. . . . . . 7 ⊢ (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
14		relexp1g 14982	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) = 𝑟)
15		1nn0 12447	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
16	10	ov2ssiunov2 44148	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
17	3, 15, 16	mp3an23 1456	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
18	14, 17	eqsstrrd 3958	. . . . . . 7 ⊢ (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
19		nn0uz 12820	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
20	10	iunrelexpuztr 44167	. . . . . . . 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 1456	. . . . . . 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 6848	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V
23		sseq2 3949	. . . . . . . . . . 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 3949	. . . . . . . . . . 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 5814	. . . . . . . . . . . 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 3956	. . . . . . . . . . 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 1439	. . . . . . . . . 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 1929	. . . . . . . 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 3615	. . . . . . . 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 588	. . . . . . 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 1344	. . . . . 6 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
34		intss1 4906	. . . . . 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 3434	. . . . . . . . 9 ⊢ 𝑠 ∈ V
37		sseq2 3949	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38		sseq2 3949	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ 𝑠))
39		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
40	39, 39	coeq12d 5814	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
41	40, 39	sseq12d 3956	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
42	37, 38, 41	3anbi123d 1439	. . . . . . . . 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 3623	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ ℕ0 = ℕ0
45	10	iunrelexpmin2 44160	. . . . . . . . . 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 692	. . . . . . . . 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 2197	. . . . . . . 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 3129	. . . . . 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 4907	. . . . . 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 3940	. . . 4 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
53		oveq1 7368	. . . . . 6 ⊢ (𝑎 = 𝑟 → (𝑎↑𝑟𝑛) = (𝑟↑𝑟𝑛))
54	53	iuneq2d 4965	. . . . 5 ⊢ (𝑎 = 𝑟 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
55		eqid 2737	. . . . 5 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))
56		ovex 7394	. . . . . 6 ⊢ (𝑟↑𝑟𝑛) ∈ V
57	3, 56	iunex 7915	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) ∈ V
58	54, 55, 57	fvmpt 6942	. . . 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 5181	. 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 206   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3430   ∪ cun 3888   ⊆ wss 3890  ∩ cint 4890  ∪ ciun 4934   ↦ cmpt 5167   I cid 5519  dom cdm 5625  ran crn 5626   ↾ cres 5627   ∘ ccom 5629  ‘cfv 6493  (class class class)co 7361  0cc0 11032  1c1 11033  ℕ₀cn0 12431  ℤ_≥cuz 12782  t*crtcl 14942  ↑_𝑟crelexp 14975

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-n0 12432  df-z 12519  df-uz 12783  df-seq 13958  df-rtrcl 14944  df-relexp 14976

This theorem is referenced by:  brfvrtrcld  44182  fvrtrcllb0d  44183  fvrtrcllb0da  44184  fvrtrcllb1d  44185  dfrtrcl4  44186  corcltrcl  44187  cotrclrcl  44190