Theorem dfnns2 28380

Description: Alternate definition of the positive surreal integers. Compare df-nn 12294. (Contributed by Scott Fenton, 6-Aug-2025.)

Assertion

Ref	Expression
dfnns2	⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)

Proof of Theorem dfnns2

Dummy variables 𝑖 𝑗 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnns 28361	. . . 4 ⊢ (𝑖 ∈ ℕs ↔ (𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ))
2		df-ne 2947	. . . . . . 7 ⊢ (𝑖 ≠ 0s ↔ ¬ 𝑖 = 0s )
3		n0s0suc 28363	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ0s → (𝑖 = 0s ∨ ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
4	3	ord 863	. . . . . . 7 ⊢ (𝑖 ∈ ℕ0s → (¬ 𝑖 = 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
5	2, 4	biimtrid 242	. . . . . 6 ⊢ (𝑖 ∈ ℕ0s → (𝑖 ≠ 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
6	5	imp 406	. . . . 5 ⊢ ((𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ) → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s ))
7		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0s → (𝑖 +s 1s ) = ( 0s +s 1s ))
8		1sno 27890	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
9		addslid 28019	. . . . . . . . . . . . . 14 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
10	8, 9	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ( 0s +s 1s ) = 1s
11	7, 10	eqtrdi 2796	. . . . . . . . . . . 12 ⊢ (𝑖 = 0s → (𝑖 +s 1s ) = 1s )
12	11	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑖 = 0s → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ))
13	12	rexbidv 3185	. . . . . . . . . 10 ⊢ (𝑖 = 0s → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ))
14		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝑖 +s 1s ) = (𝑘 +s 1s ))
15	14	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s )))
16	15	rexbidv 3185	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s )))
17		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑘 +s 1s ) → (𝑖 +s 1s ) = ((𝑘 +s 1s ) +s 1s ))
18	17	eqeq2d 2751	. . . . . . . . . . . 12 ⊢ (𝑖 = (𝑘 +s 1s ) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = ((𝑘 +s 1s ) +s 1s )))
19	18	rexbidv 3185	. . . . . . . . . . 11 ⊢ (𝑖 = (𝑘 +s 1s ) → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = ((𝑘 +s 1s ) +s 1s )))
20		fveqeq2 6929	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = ((𝑘 +s 1s ) +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
21	20	cbvrexvw 3244	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = ((𝑘 +s 1s ) +s 1s ) ↔ ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s ))
22	19, 21	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑖 = (𝑘 +s 1s ) → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
23		oveq1 7455	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑖 +s 1s ) = (𝑗 +s 1s ))
24	23	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s )))
25	24	rexbidv 3185	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑖 +s 1s ) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s )))
26		peano1 7927	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
27		1nns 28370	. . . . . . . . . . . 12 ⊢ 1s ∈ ℕs
28		fr0g 8492	. . . . . . . . . . . 12 ⊢ ( 1s ∈ ℕs → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s )
29	27, 28	ax-mp 5	. . . . . . . . . . 11 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s
30		fveqeq2 6929	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ))
31	30	rspcev 3635	. . . . . . . . . . 11 ⊢ ((∅ ∈ ω ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ) → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s )
32	26, 29, 31	mp2an 691	. . . . . . . . . 10 ⊢ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s
33		fveqeq2 6929	. . . . . . . . . . . . . 14 ⊢ (𝑧 = suc 𝑦 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s )))
34		peano2 7929	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
35		ovex 7481	. . . . . . . . . . . . . . 15 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ∈ V
36		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
37		oveq1 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
38		oveq1 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) → (𝑧 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ))
39	36, 37, 38	frsucmpt2 8496	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ))
40	35, 39	mpan2 690	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑦) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ))
41	33, 34, 40	rspcedvdw 3638	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ))
42	41	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ ℕ0s ∧ 𝑦 ∈ ω) → ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ))
43		oveq1 7455	. . . . . . . . . . . . . 14 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s ) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) = ((𝑘 +s 1s ) +s 1s ))
44	43	eqeq2d 2751	. . . . . . . . . . . . 13 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s ) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
45	44	rexbidv 3185	. . . . . . . . . . . 12 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s ) → (∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ↔ ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
46	42, 45	syl5ibcom 245	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℕ0s ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s ) → ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
47	46	rexlimdva 3161	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0s → (∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑘 +s 1s ) → ∃𝑧 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑧) = ((𝑘 +s 1s ) +s 1s )))
48	13, 16, 22, 25, 32, 47	n0sind 28355	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ0s → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
49		frfnom 8491	. . . . . . . . . 10 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω
50		fvelrnb 6982	. . . . . . . . . 10 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω → ((𝑗 +s 1s ) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s )))
51	49, 50	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑗 +s 1s ) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ↔ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
52	48, 51	sylibr 234	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω))
53		df-ima 5713	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
54	52, 53	eleqtrrdi 2855	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
55		eleq1 2832	. . . . . . 7 ⊢ (𝑖 = (𝑗 +s 1s ) → (𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ↔ (𝑗 +s 1s ) ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)))
56	54, 55	syl5ibrcom 247	. . . . . 6 ⊢ (𝑗 ∈ ℕ0s → (𝑖 = (𝑗 +s 1s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)))
57	56	rexlimiv 3154	. . . . 5 ⊢ (∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
58	6, 57	syl 17	. . . 4 ⊢ ((𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
59	1, 58	sylbi 217	. . 3 ⊢ (𝑖 ∈ ℕs → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
60	59	ssriv 4012	. 2 ⊢ ℕs ⊆ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
61		fveq2 6920	. . . . . . 7 ⊢ (𝑘 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅))
62	61	eleq1d 2829	. . . . . 6 ⊢ (𝑘 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs))
63		fveq2 6920	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗))
64	63	eleq1d 2829	. . . . . 6 ⊢ (𝑘 = 𝑗 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) ∈ ℕs))
65		fveq2 6920	. . . . . . 7 ⊢ (𝑘 = suc 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗))
66	65	eleq1d 2829	. . . . . 6 ⊢ (𝑘 = suc 𝑗 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗) ∈ ℕs))
67		fveq2 6920	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖))
68	67	eleq1d 2829	. . . . . 6 ⊢ (𝑘 = 𝑖 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs))
69	29, 27	eqeltri 2840	. . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs
70		peano2nns 28371	. . . . . . 7 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) ∈ ℕs → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ ℕs)
71		ovex 7481	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V
72		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
73		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑦 = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) → (𝑦 +s 1s ) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ))
74	36, 72, 73	frsucmpt2 8496	. . . . . . . . 9 ⊢ ((𝑗 ∈ ω ∧ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ))
75	71, 74	mpan2 690	. . . . . . . 8 ⊢ (𝑗 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗) = (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ))
76	75	eleq1d 2829	. . . . . . 7 ⊢ (𝑗 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗) ∈ ℕs ↔ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ ℕs))
77	70, 76	imbitrrid 246	. . . . . 6 ⊢ (𝑗 ∈ ω → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) ∈ ℕs → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗) ∈ ℕs))
78	62, 64, 66, 68, 69, 77	finds 7936	. . . . 5 ⊢ (𝑖 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs)
79	78	rgen 3069	. . . 4 ⊢ ∀𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs
80		fnfvrnss 7155	. . . 4 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω ∧ ∀𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs) → ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ⊆ ℕs)
81	49, 79, 80	mp2an 691	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ⊆ ℕs
82	53, 81	eqsstri 4043	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ⊆ ℕs
83	60, 82	eqssi 4025	1 ⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   ↦ cmpt 5249  ran crn 5701   ↾ cres 5702   “ cima 5703  suc csuc 6397   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  ωcom 7903  reccrdg 8465   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  ℕ_0scnn0s 28336  ℕ_scnns 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011  df-n0s 28338  df-nns 28339

This theorem is referenced by:  nnsind  28381  expsp1  28430