Theorem dfnns2 28318

Description: Alternate definition of the positive surreal integers. Compare df-nn 12246. (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 28289	. . . 4 ⊢ (𝑖 ∈ ℕs ↔ (𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ))
2		df-ne 2934	. . . . . . 7 ⊢ (𝑖 ≠ 0s ↔ ¬ 𝑖 = 0s )
3		n0s0suc 28291	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ0s → (𝑖 = 0s ∨ ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
4	3	ord 864	. . . . . . 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 7417	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0s → (𝑖 +s 1s ) = ( 0s +s 1s ))
8		1sno 27796	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
9		addslid 27932	. . . . . . . . . . . . . 14 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
10	8, 9	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ( 0s +s 1s ) = 1s
11	7, 10	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ (𝑖 = 0s → (𝑖 +s 1s ) = 1s )
12	11	eqeq2d 2747	. . . . . . . . . . 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 3165	. . . . . . . . . 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 7417	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝑖 +s 1s ) = (𝑘 +s 1s ))
15	14	eqeq2d 2747	. . . . . . . . . . 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 3165	. . . . . . . . . 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 7417	. . . . . . . . . . . . 13 ⊢ (𝑖 = (𝑘 +s 1s ) → (𝑖 +s 1s ) = ((𝑘 +s 1s ) +s 1s ))
18	17	eqeq2d 2747	. . . . . . . . . . . 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 3165	. . . . . . . . . . 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 6890	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 7417	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑖 +s 1s ) = (𝑗 +s 1s ))
24	23	eqeq2d 2747	. . . . . . . . . . 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 3165	. . . . . . . . . 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 7889	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
27		1nns 28298	. . . . . . . . . . . 12 ⊢ 1s ∈ ℕs
28		fr0g 8455	. . . . . . . . . . . 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 6890	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ))
31	30	rspcev 3606	. . . . . . . . . . 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 692	. . . . . . . . . 10 ⊢ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s
33		fveqeq2 6890	. . . . . . . . . . . . . 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 7891	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
35		ovex 7443	. . . . . . . . . . . . . . 15 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ∈ V
36		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
37		oveq1 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
38		oveq1 7417	. . . . . . . . . . . . . . . 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 8459	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . 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 3609	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . 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 3165	. . . . . . . . . . . 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 3142	. . . . . . . . . 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 28282	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ0s → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
49		frfnom 8454	. . . . . . . . . 10 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω
50		fvelrnb 6944	. . . . . . . . . 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 5672	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
54	52, 53	eleqtrrdi 2846	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
55		eleq1 2823	. . . . . . 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 3135	. . . . 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 3967	. 2 ⊢ ℕs ⊆ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
61		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅))
62	61	eleq1d 2820	. . . . . 6 ⊢ (𝑘 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs))
63		fveq2 6881	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗))
64	63	eleq1d 2820	. . . . . 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 6881	. . . . . . 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 2820	. . . . . 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 6881	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖))
68	67	eleq1d 2820	. . . . . 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 2831	. . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs
70		peano2nns 28299	. . . . . . 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 7443	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V
72		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
73		oveq1 7417	. . . . . . . . . 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 8459	. . . . . . . . 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 691	. . . . . . . 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 2820	. . . . . . 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 7897	. . . . 5 ⊢ (𝑖 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs)
79	78	rgen 3054	. . . 4 ⊢ ∀𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs
80		fnfvrnss 7116	. . . 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 692	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ⊆ ℕs
82	53, 81	eqsstri 4010	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ⊆ ℕs
83	60, 82	eqssi 3980	1 ⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ⊆ wss 3931  ∅c0 4313   ↦ cmpt 5206  ran crn 5660   ↾ cres 5661   “ cima 5662  suc csuc 6359   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410  ωcom 7866  reccrdg 8428   No csur 27608   0_s c0s 27791   1_s c1s 27792   +_s cadds 27923  ℕ_0scnn0s 28263  ℕ_scnns 28264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-nadd 8683  df-no 27611  df-slt 27612  df-bday 27613  df-sle 27714  df-sslt 27750  df-scut 27752  df-0s 27793  df-1s 27794  df-made 27812  df-old 27813  df-left 27815  df-right 27816  df-norec2 27913  df-adds 27924  df-n0s 28265  df-nns 28266

This theorem is referenced by:  nnsind  28319  expsp1  28372