Theorem rtrclreclem4 15034

Description: The reflexive, transitive closure of 𝑅 is the smallest reflexive, transitive relation which contains 𝑅 and the identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)

Hypothesis

Ref	Expression
rtrclreclem.1	⊢ (𝜑 → Rel 𝑅)

Assertion

Ref	Expression
rtrclreclem4	⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))

Distinct variable group:   𝜑,𝑠

Allowed substitution hint:   𝑅(𝑠)

Proof of Theorem rtrclreclem4

Dummy variables 𝑛 𝑖 𝑚 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2728	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)))
2		oveq1 7421	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 5020	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
4	3	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ 𝑟 = 𝑅) → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
5		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
6		nn0ex 12502	. . . . . . . . 9 ⊢ ℕ0 ∈ V
7		ovex 7447	. . . . . . . . 9 ⊢ (𝑅↑𝑟𝑛) ∈ V
8	6, 7	iunex 7966	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V
9	8	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ∈ V)
10	1, 4, 5, 9	fvmptd 7006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
11		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑖 ∈ ℕ0 ↔ 0 ∈ ℕ0))
12	11	anbi1d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 0 → ((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
13		oveq2 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟0))
14	13	sseq1d 4009	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 0 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟0) ⊆ 𝑠))
15	12, 14	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑖) ⊆ 𝑠) ↔ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟0) ⊆ 𝑠)))
16		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝑖 ∈ ℕ0 ↔ 𝑚 ∈ ℕ0))
17	16	anbi1d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → ((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
18		oveq2 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟𝑚))
19	18	sseq1d 4009	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑚) ⊆ 𝑠))
20	17, 19	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑚 → (((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑖) ⊆ 𝑠) ↔ ((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠)))
21		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ ℕ0 ↔ (𝑚 + 1) ∈ ℕ0))
22	21	anbi1d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑚 + 1) → ((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
23		oveq2 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑚 + 1) → (𝑅↑𝑟𝑖) = (𝑅↑𝑟(𝑚 + 1)))
24	23	sseq1d 4009	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑚 + 1) → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
25	22, 24	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑚 + 1) → (((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑖) ⊆ 𝑠) ↔ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)))
26		eleq1 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑛 → (𝑖 ∈ ℕ0 ↔ 𝑛 ∈ ℕ0))
27	26	anbi1d 629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑛 → ((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ↔ (𝑛 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))))))
28		oveq2 7422	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑛 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟𝑛))
29	28	sseq1d 4009	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑛 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑛) ⊆ 𝑠))
30	27, 29	imbi12d 344	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 𝑛 → (((𝑖 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑖) ⊆ 𝑠) ↔ ((𝑛 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑛) ⊆ 𝑠)))
31		simprll 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝜑)
32		rtrclreclem.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → Rel 𝑅)
33	31, 32	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → Rel 𝑅)
34		simprlr 779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝑅 ∈ V)
35	33, 34	relexp0d 14997	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
36		relfld 6273	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
37	33, 36	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
38		simprrr 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
39	38	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
40		reseq2 5974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	40	sseq1d 4009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ ∪ ∪ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
42	39, 41	imbitrrid 245	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑠))
43	37, 42	mpcom 38	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑠)
44	35, 43	eqsstrd 4016	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟0) ⊆ 𝑠)
45		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → 𝑚 ∈ ℕ0)
46	45	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → 𝑚 ∈ ℕ0)
47	46	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑚 ∈ ℕ0)
48	47	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑚 ∈ ℕ0)
49		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝜑 ∧ 𝑅 ∈ V))
50		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑠 ∘ 𝑠) ⊆ 𝑠)
51		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → 𝑅 ⊆ 𝑠)
52	51	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝑅 ⊆ 𝑠)
53		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
54	53	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
55	54	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
56	50, 52, 55	jca32 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))
57	48, 49, 56	jca32 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))))
58		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠))
59	58	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠))
60	59	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠))
61	60	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → ((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠))
62	57, 61	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅↑𝑟𝑚) ⊆ 𝑠)
63		simprll 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → 𝜑)
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝜑)
65	64, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → Rel 𝑅)
66	48	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑚 ∈ ℕ0)
67	65, 66	relexpsucrd 15006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅↑𝑟(𝑚 + 1)) = ((𝑅↑𝑟𝑚) ∘ 𝑅))
68	52	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → 𝑅 ⊆ 𝑠)
69		coss2 5853	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑅 ⊆ 𝑠 → ((𝑅↑𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅↑𝑟𝑚) ∘ 𝑠))
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅↑𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅↑𝑟𝑚) ∘ 𝑠))
71		coss1 5852	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅↑𝑟𝑚) ⊆ 𝑠 → ((𝑅↑𝑟𝑚) ∘ 𝑠) ⊆ (𝑠 ∘ 𝑠))
72	71, 50	sylan9ss 3991	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅↑𝑟𝑚) ∘ 𝑠) ⊆ 𝑠)
73	70, 72	sstrd 3988	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → ((𝑅↑𝑟𝑚) ∘ 𝑅) ⊆ 𝑠)
74	67, 73	eqsstrd 4016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑅↑𝑟𝑚) ⊆ 𝑠 ∧ ((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)
75	62, 74	mpancom 687	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)
76	75	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
77	76	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)))) → ((𝜑 ∧ 𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)))
78	77	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ⊆ 𝑠 ∧ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))))
79	78	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))))
80	79	impcom 407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ ((𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝜑 ∧ 𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)))
81	80	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝜑 ∧ 𝑅 ∈ V) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)))
82	81	impcom 407	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
83	82	anassrs 467	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → ((𝑚 + 1) ∈ ℕ0 → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
84	83	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)
85	84	anassrs 467	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) ∧ (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0)) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)
86	85	expcom 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
87	86	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → (((𝑚 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑚) ⊆ 𝑠) → (((𝑚 + 1) ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠)))
88	15, 20, 25, 30, 44, 87	nn0ind 12681	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑛) ⊆ 𝑠))
89	88	anabsi5 668	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑛) ⊆ 𝑠)
90	89	expcom 413	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅↑𝑟𝑛) ⊆ 𝑠))
91	90	ralrimiv 3140	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
92		iunss 5042	. . . . . . . . . . . . 13 ⊢ (∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
93	91, 92	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
94	93	expcom 413	. . . . . . . . . . 11 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
95	94	expcom 413	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)))
96	95	expcom 413	. . . . . . . . 9 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅 ⊆ 𝑠 → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))))
97	96	3imp1 1345	. . . . . . . 8 ⊢ (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ∧ (𝜑 ∧ 𝑅 ∈ V)) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
98	97	expcom 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
99		sseq1 4003	. . . . . . . 8 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠 ↔ ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
100	99	imbi2d 340	. . . . . . 7 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠) ↔ ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)))
101	98, 100	imbitrrid 245	. . . . . 6 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) → ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠)))
102	10, 101	mpcom 38	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠))
103		df-rtrclrec 15029	. . . . . 6 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
104		fveq1 6890	. . . . . . . . 9 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
105	104	sseq1d 4009	. . . . . . . 8 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠))
106	105	imbi2d 340	. . . . . . 7 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠) ↔ ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠)))
107	106	imbi2d 340	. . . . . 6 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠))))
108	103, 107	ax-mp 5	. . . . 5 ⊢ (((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) ↔ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠)))
109	102, 108	mpbir 230	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
110	109	ex 412	. . 3 ⊢ (𝜑 → (𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)))
111		fvprc 6883	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
112		0ss 4392	. . . . 5 ⊢ ∅ ⊆ 𝑠
113	111, 112	eqsstrdi 4032	. . . 4 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) ⊆ 𝑠)
114	113	a1d 25	. . 3 ⊢ (¬ 𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
115	110, 114	pm2.61d1 180	. 2 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
116	115	alrimiv 1923	1 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3056  Vcvv 3469   ∪ cun 3942   ⊆ wss 3944  ∅c0 4318  ∪ cuni 4903  ∪ ciun 4991   ↦ cmpt 5225   I cid 5569  dom cdm 5672  ran crn 5673   ↾ cres 5674   ∘ ccom 5676  Rel wrel 5677  ‘cfv 6542  (class class class)co 7414  0cc0 11132  1c1 11133   + caddc 11135  ℕ₀cn0 12496  ↑_𝑟crelexp 14992  t*reccrtrcl 15028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-n0 12497  df-z 12583  df-uz 12847  df-seq 13993  df-relexp 14993  df-rtrclrec 15029

This theorem is referenced by:  dfrtrcl2  15035