Theorem dfnns2 28389

Description: Alternate definition of the positive surreal integers. Compare df-nn 12173. (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 28357	. . . 4 ⊢ (𝑖 ∈ ℕs ↔ (𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ))
2		df-ne 2936	. . . . . . 7 ⊢ (𝑖 ≠ 0s ↔ ¬ 𝑖 = 0s )
3		n0s0suc 28359	. . . . . . . 8 ⊢ (𝑖 ∈ ℕ0s → (𝑖 = 0s ∨ ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
4	3	ord 870	. . . . . . 7 ⊢ (𝑖 ∈ ℕ0s → (¬ 𝑖 = 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
5	2, 4	biimtrid 243	. . . . . 6 ⊢ (𝑖 ∈ ℕ0s → (𝑖 ≠ 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
6	5	imp 407	. . . . 5 ⊢ ((𝑖 ∈ ℕ0s ∧ 𝑖 ≠ 0s ) → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s ))
7		oveq1 7370	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0s → (𝑖 +s 1s ) = ( 0s +s 1s ))
8		1no 27827	. . . . . . . . . . . . . 14 ⊢ 1s ∈ No
9		addslid 27985	. . . . . . . . . . . . . 14 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
10	8, 9	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ( 0s +s 1s ) = 1s
11	7, 10	eqtrdi 2791	. . . . . . . . . . . 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 3164	. . . . . . . . . 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 7370	. . . . . . . . . . . 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 3164	. . . . . . . . . 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 7370	. . . . . . . . . . . . 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 3164	. . . . . . . . . . 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 6843	. . . . . . . . . . . 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 3219	. . . . . . . . . . 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 288	. . . . . . . . . 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 7370	. . . . . . . . . . . 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 3164	. . . . . . . . . 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 7836	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
27		1nns 28366	. . . . . . . . . . . 12 ⊢ 1s ∈ ℕs
28		fr0g 8372	. . . . . . . . . . . 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 6843	. . . . . . . . . . . 12 ⊢ (𝑦 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ))
31	30	rspcev 3567	. . . . . . . . . . 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 698	. . . . . . . . . 10 ⊢ ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s
33		fveqeq2 6843	. . . . . . . . . . . . . 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 7837	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
35		ovex 7396	. . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
38		oveq1 7370	. . . . . . . . . . . . . . . 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 8376	. . . . . . . . . . . . . . 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 697	. . . . . . . . . . . . . 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 3570	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 7370	. . . . . . . . . . . . . 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 3164	. . . . . . . . . . . 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 246	. . . . . . . . . . 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 3141	. . . . . . . . . 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 28350	. . . . . . . . 9 ⊢ (𝑗 ∈ ℕ0s → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
49		frfnom 8371	. . . . . . . . . 10 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω
50		fvelrnb 6894	. . . . . . . . . 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 235	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω))
53		df-ima 5638	. . . . . . . 8 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
54	52, 53	eleqtrrdi 2851	. . . . . . 7 ⊢ (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
55		eleq1 2828	. . . . . . 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 248	. . . . . 6 ⊢ (𝑗 ∈ ℕ0s → (𝑖 = (𝑗 +s 1s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)))
57	56	rexlimiv 3134	. . . . 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 218	. . 3 ⊢ (𝑖 ∈ ℕs → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
60	59	ssriv 3926	. 2 ⊢ ℕs ⊆ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
61		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅))
62	61	eleq1d 2825	. . . . . 6 ⊢ (𝑘 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs))
63		fveq2 6834	. . . . . . 7 ⊢ (𝑘 = 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗))
64	63	eleq1d 2825	. . . . . 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 6834	. . . . . . 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 2825	. . . . . 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 6834	. . . . . . 7 ⊢ (𝑘 = 𝑖 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖))
68	67	eleq1d 2825	. . . . . 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 2836	. . . . . 6 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs
70		peano2nns 28367	. . . . . . 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 7396	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V
72		oveq1 7370	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
73		oveq1 7370	. . . . . . . . . 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 8376	. . . . . . . . 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 697	. . . . . . . 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 2825	. . . . . . 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 247	. . . . . 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 7843	. . . . 5 ⊢ (𝑖 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs)
79	78	rgen 3056	. . . 4 ⊢ ∀𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs
80		fnfvrnss 7069	. . . 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 698	. . 3 ⊢ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ⊆ ℕs
82	53, 81	eqsstri 3968	. 2 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ⊆ ℕs
83	60, 82	eqssi 3938	1 ⊢ ℕs = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  Vcvv 3432   ⊆ wss 3890  ∅c0 4268   ↦ cmpt 5160  ran crn 5626   ↾ cres 5627   “ cima 5628  suc csuc 6319   Fn wfn 6487  ‘cfv 6492  (class class class)co 7363  ωcom 7813  reccrdg 8345   No csur 27628   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976  ℕ_0scn0s 28329  ℕ_scnns 28330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec2 27966  df-adds 27977  df-n0s 28331  df-nns 28332

This theorem is referenced by:  nnsind  28390  expsp1  28446