Theorem dfrtrcl3 40434

Description: Reflexive-transitive closure of a relation, expressed as indexed union of powers of relations. Generalized from dfrtrcl2 14413. (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 14339	. 2 ⊢ t* = (𝑟 ∈ V ↦ ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
2		relexp0g 14373	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) = ( I ↾ (dom 𝑟 ∪ ran 𝑟)))
3		nn0ex 11891	. . . . . . . . 9 ⊢ ℕ0 ∈ V
4		0nn0 11900	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
5		oveq1 7142	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑡 → (𝑎↑𝑟𝑛) = (𝑡↑𝑟𝑛))
6	5	iuneq2d 4910	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛))
7		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑡↑𝑟𝑛) = (𝑡↑𝑟𝑘))
8	7	cbviunv 4927	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑡↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘)
9	6, 8	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑡 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
10	9	cbvmptv 5133	. . . . . . . . . 10 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑡 ∈ V ↦ ∪ 𝑘 ∈ ℕ0 (𝑡↑𝑟𝑘))
11	10	ov2ssiunov2 40401	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 0 ∈ ℕ0) → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
12	3, 4, 11	mp3an23 1450	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟0) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
13	2, 12	eqsstrrd 3954	. . . . . . 7 ⊢ (𝑟 ∈ V → ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
14		relexp1g 14377	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) = 𝑟)
15		1nn0 11901	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
16	10	ov2ssiunov2 40401	. . . . . . . . 9 ⊢ ((𝑟 ∈ V ∧ ℕ0 ∈ V ∧ 1 ∈ ℕ0) → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
17	3, 15, 16	mp3an23 1450	. . . . . . . 8 ⊢ (𝑟 ∈ V → (𝑟↑𝑟1) ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
18	14, 17	eqsstrrd 3954	. . . . . . 7 ⊢ (𝑟 ∈ V → 𝑟 ⊆ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
19		nn0uz 12268	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
20	10	iunrelexpuztr 40420	. . . . . . . 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 1450	. . . . . . 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 6658	. . . . . . . 8 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ V
23		sseq2 3941	. . . . . . . . . . 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 3941	. . . . . . . . . . 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 5699	. . . . . . . . . . . 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 3948	. . . . . . . . . . 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 1433	. . . . . . . . . 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 1928	. . . . . . . 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 3609	. . . . . . . 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 590	. . . . . . 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 1339	. . . . . 6 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) ∈ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
34		intss1 4853	. . . . . 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 3444	. . . . . . . . 9 ⊢ 𝑠 ∈ V
37		sseq2 3941	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑠))
38		sseq2 3941	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → (𝑟 ⊆ 𝑧 ↔ 𝑟 ⊆ 𝑠))
39		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠)
40	39, 39	coeq12d 5699	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠))
41	40, 39	sseq12d 3948	. . . . . . . . . 10 ⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠))
42	37, 38, 41	3anbi123d 1433	. . . . . . . . 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 3615	. . . . . . . 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 2798	. . . . . . . . . 10 ⊢ ℕ0 = ℕ0
45	10	iunrelexpmin2 40413	. . . . . . . . . 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 690	. . . . . . . . 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 2186	. . . . . . . 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 245	. . . . . . 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 3148	. . . . . 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 4854	. . . . . 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 237	. . . . 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 3932	. . . 4 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟))
53		oveq1 7142	. . . . . 6 ⊢ (𝑎 = 𝑟 → (𝑎↑𝑟𝑛) = (𝑟↑𝑟𝑛))
54	53	iuneq2d 4910	. . . . 5 ⊢ (𝑎 = 𝑟 → ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
55		eqid 2798	. . . . 5 ⊢ (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛)) = (𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))
56		ovex 7168	. . . . . 6 ⊢ (𝑟↑𝑟𝑛) ∈ V
57	3, 56	iunex 7651	. . . . 5 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) ∈ V
58	54, 55, 57	fvmpt 6745	. . . 4 ⊢ (𝑟 ∈ V → ((𝑎 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑎↑𝑟𝑛))‘𝑟) = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
59	52, 58	eqtrd 2833	. . 3 ⊢ (𝑟 ∈ V → ∩ {𝑧 ∣ (( I ↾ (dom 𝑟 ∪ ran 𝑟)) ⊆ 𝑧 ∧ 𝑟 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
60	59	mpteq2ia 5121	. 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 2821	1 ⊢ t* = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  Vcvv 3441   ∪ cun 3879   ⊆ wss 3881  ∩ cint 4838  ∪ ciun 4881   ↦ cmpt 5110   I cid 5424  dom cdm 5519  ran crn 5520   ↾ cres 5521   ∘ ccom 5523  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527  ℕ₀cn0 11885  ℤ_≥cuz 12231  t*crtcl 14337  ↑_𝑟crelexp 14370

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-seq 13365  df-rtrcl 14339  df-relexp 14371

This theorem is referenced by:  brfvrtrcld  40435  fvrtrcllb0d  40436  fvrtrcllb0da  40437  fvrtrcllb1d  40438  dfrtrcl4  40439  corcltrcl  40440  cotrclrcl  40443