Theorem rtrclreclem4 15134

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 2761	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)))
2		oveq1 7420	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑟↑𝑟𝑛) = (𝑅↑𝑟𝑛))
3	2	iuneq2d 4981	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
4	3	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ 𝑟 = 𝑅) → ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
5		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
6		nn0ex 12534	. . . . . . . . 9 ⊢ ℕ0 ∈ V
7		ovex 7446	. . . . . . . . 9 ⊢ (𝑅↑𝑟𝑛) ∈ V
8	6, 7	iunex 7965	. . . . . . . 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 6994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) = ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛))
11		eleq1 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑖 ∈ ℕ0 ↔ 0 ∈ ℕ0))
12	11	anbi1d 643	. . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 0 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟0))
14	13	sseq1d 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 0 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟0) ⊆ 𝑠))
15	12, 14	imbi12d 347	. . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝑖 ∈ ℕ0 ↔ 𝑚 ∈ ℕ0))
17	16	anbi1d 643	. . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑚 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟𝑚))
19	18	sseq1d 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑚 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑚) ⊆ 𝑠))
20	17, 19	imbi12d 347	. . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑚 + 1) → (𝑖 ∈ ℕ0 ↔ (𝑚 + 1) ∈ ℕ0))
22	21	anbi1d 643	. . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = (𝑚 + 1) → (𝑅↑𝑟𝑖) = (𝑅↑𝑟(𝑚 + 1)))
24	23	sseq1d 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑚 + 1) → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟(𝑚 + 1)) ⊆ 𝑠))
25	22, 24	imbi12d 347	. . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑛 → (𝑖 ∈ ℕ0 ↔ 𝑛 ∈ ℕ0))
27	26	anbi1d 643	. . . . . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑛 → (𝑅↑𝑟𝑖) = (𝑅↑𝑟𝑛))
29	28	sseq1d 3962	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝑛 → ((𝑅↑𝑟𝑖) ⊆ 𝑠 ↔ (𝑅↑𝑟𝑛) ⊆ 𝑠))
30	27, 29	imbi12d 347	. . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → Rel 𝑅)
34		simprlr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → 𝑅 ∈ V)
35	33, 34	relexp0d 15097	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
36		relfld 6272	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
37	33, 36	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
38		simprrr 794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)
39	38	adantl 487	. . . . . . . . . . . . . . . . . . . 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 5967	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
41	40	sseq1d 3962	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (( I ↾ ∪ ∪ 𝑅) ⊆ 𝑠 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))
42	39, 41	imbitrrid 249	. . . . . . . . . . . . . . . . . . 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 39	. . . . . . . . . . . . . . . . . 18 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑠)
44	35, 43	eqsstrd 3965	. . . . . . . . . . . . . . . . 17 ⊢ ((0 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟0) ⊆ 𝑠)
45		simprrr 794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15106	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑅 ⊆ 𝑠 → ((𝑅↑𝑟𝑚) ∘ 𝑅) ⊆ ((𝑅↑𝑟𝑚) ∘ 𝑠))
70	68, 69	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑅↑𝑟𝑚) ⊆ 𝑠 → ((𝑅↑𝑟𝑚) ∘ 𝑠) ⊆ (𝑠 ∘ 𝑠))
72	71, 50	sylan9ss 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 701	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . 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 12716	. . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0 ∧ ((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)))) → (𝑅↑𝑟𝑛) ⊆ 𝑠)
90	89	expcom 419	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → (𝑛 ∈ ℕ0 → (𝑅↑𝑟𝑛) ⊆ 𝑠))
91	90	ralrimiv 3153	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
92		iunss 5003	. . . . . . . . . . . . 13 ⊢ (∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠 ↔ ∀𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
93	91, 92	sylibr 237	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑅 ∈ V) ∧ ((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠))) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
94	93	expcom 419	. . . . . . . . . . 11 ⊢ (((𝑠 ∘ 𝑠) ⊆ 𝑠 ∧ (𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
95	94	expcom 419	. . . . . . . . . 10 ⊢ ((𝑅 ⊆ 𝑠 ∧ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠) → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)))
96	95	expcom 419	. . . . . . . . 9 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 → (𝑅 ⊆ 𝑠 → ((𝑠 ∘ 𝑠) ⊆ 𝑠 → ((𝜑 ∧ 𝑅 ∈ V) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))))
97	96	3imp1 1366	. . . . . . . 8 ⊢ (((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) ∧ (𝜑 ∧ 𝑅 ∈ V)) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠)
98	97	expcom 419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → ∪ 𝑛 ∈ ℕ0 (𝑅↑𝑟𝑛) ⊆ 𝑠))
99		sseq1 3956	. . . . . . . 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 343	. . . . . . 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 249	. . . . . 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 39	. . . . 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 15129	. . . . . 6 ⊢ t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))
104		fveq1 6877	. . . . . . . . 9 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅))
105	104	sseq1d 3962	. . . . . . . 8 ⊢ (t*rec = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛)) → ((t*rec‘𝑅) ⊆ 𝑠 ↔ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ ℕ0 (𝑟↑𝑟𝑛))‘𝑅) ⊆ 𝑠))
106	105	imbi2d 343	. . . . . . 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 343	. . . . . 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 234	. . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
110	109	ex 418	. . 3 ⊢ (𝜑 → (𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)))
111		fvprc 6870	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) = ∅)
112		0ss 4350	. . . . 5 ⊢ ∅ ⊆ 𝑠
113	111, 112	eqsstrdi 3975	. . . 4 ⊢ (¬ 𝑅 ∈ V → (t*rec‘𝑅) ⊆ 𝑠)
114	113	a1d 26	. . 3 ⊢ (¬ 𝑅 ∈ V → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
115	110, 114	pm2.61d1 182	. 2 ⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))
116	115	alrimiv 1960	1 ⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867  ∪ ciun 4951   ↦ cmpt 5186   I cid 5549  dom cdm 5655  ran crn 5656   ↾ cres 5657   ∘ ccom 5659  Rel wrel 5660  ‘cfv 6533  (class class class)co 7413  0cc0 11124  1c1 11125   + caddc 11127  ℕ₀cn0 12528  ↑_𝑟crelexp 15092  t*reccrtrcl 15128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-n0 12529  df-z 12616  df-uz 12888  df-seq 14066  df-relexp 15093  df-rtrclrec 15129

This theorem is used by:  dfrtrcl2  15135