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Theorem dfnns2 28363
Description: Alternate definition of the positive surreal integers. Compare df-nn 12268. (Contributed by Scott Fenton, 6-Aug-2025.)
Assertion
Ref Expression
dfnns2 s = (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.
StepHypRef Expression
1 elnns 28344 . . . 4 (𝑖 ∈ ℕs ↔ (𝑖 ∈ ℕ0s𝑖 ≠ 0s ))
2 df-ne 2940 . . . . . . 7 (𝑖 ≠ 0s ↔ ¬ 𝑖 = 0s )
3 n0s0suc 28346 . . . . . . . 8 (𝑖 ∈ ℕ0s → (𝑖 = 0s ∨ ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
43ord 864 . . . . . . 7 (𝑖 ∈ ℕ0s → (¬ 𝑖 = 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
52, 4biimtrid 242 . . . . . 6 (𝑖 ∈ ℕ0s → (𝑖 ≠ 0s → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s )))
65imp 406 . . . . 5 ((𝑖 ∈ ℕ0s𝑖 ≠ 0s ) → ∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s ))
7 oveq1 7439 . . . . . . . . . . . . 13 (𝑖 = 0s → (𝑖 +s 1s ) = ( 0s +s 1s ))
8 1sno 27873 . . . . . . . . . . . . . 14 1s No
9 addslid 28002 . . . . . . . . . . . . . 14 ( 1s No → ( 0s +s 1s ) = 1s )
108, 9ax-mp 5 . . . . . . . . . . . . 13 ( 0s +s 1s ) = 1s
117, 10eqtrdi 2792 . . . . . . . . . . . 12 (𝑖 = 0s → (𝑖 +s 1s ) = 1s )
1211eqeq2d 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 ))
1312rexbidv 3178 . . . . . . . . . 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 7439 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑖 +s 1s ) = (𝑘 +s 1s ))
1514eqeq2d 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 )))
1615rexbidv 3178 . . . . . . . . . 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 7439 . . . . . . . . . . . . 13 (𝑖 = (𝑘 +s 1s ) → (𝑖 +s 1s ) = ((𝑘 +s 1s ) +s 1s ))
1817eqeq2d 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 )))
1918rexbidv 3178 . . . . . . . . . . 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 6914 . . . . . . . . . . . 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 )))
2120cbvrexvw 3237 . . . . . . . . . . 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 ))
2219, 21bitrdi 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 7439 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 +s 1s ) = (𝑗 +s 1s ))
2423eqeq2d 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 )))
2524rexbidv 3178 . . . . . . . . . 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 7911 . . . . . . . . . . 11 ∅ ∈ ω
27 1nns 28353 . . . . . . . . . . . 12 1s ∈ ℕs
28 fr0g 8477 . . . . . . . . . . . 12 ( 1s ∈ ℕs → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s )
2927, 28ax-mp 5 . . . . . . . . . . 11 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s
30 fveqeq2 6914 . . . . . . . . . . . 12 (𝑦 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ))
3130rspcev 3621 . . . . . . . . . . 11 ((∅ ∈ ω ∧ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ) → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s )
3226, 29, 31mp2an 692 . . . . . . . . . 10 𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s
33 fveqeq2 6914 . . . . . . . . . . . . . 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 7913 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
35 ovex 7465 . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
38 oveq1 7439 . . . . . . . . . . . . . . . 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 ))
3936, 37, 38frsucmpt2 8481 . . . . . . . . . . . . . . 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 ))
4035, 39mpan2 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 ))
4133, 34, 40rspcedvdw 3624 . . . . . . . . . . . . 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 ))
4241adantl 481 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ0s𝑦 ∈ 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 7439 . . . . . . . . . . . . . 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 ))
4443eqeq2d 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 )))
4544rexbidv 3178 . . . . . . . . . . . 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 )))
4642, 45syl5ibcom 245 . . . . . . . . . . 11 ((𝑘 ∈ ℕ0s𝑦 ∈ 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 )))
4746rexlimdva 3154 . . . . . . . . . 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 )))
4813, 16, 22, 25, 32, 47n0sind 28338 . . . . . . . . 9 (𝑗 ∈ ℕ0s → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
49 frfnom 8476 . . . . . . . . . 10 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω
50 fvelrnb 6968 . . . . . . . . . 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 )))
5149, 50ax-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 ))
5248, 51sylibr 234 . . . . . . . 8 (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω))
53 df-ima 5697 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
5452, 53eleqtrrdi 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)))
5654, 55syl5ibrcom 247 . . . . . 6 (𝑗 ∈ ℕ0s → (𝑖 = (𝑗 +s 1s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)))
5756rexlimiv 3147 . . . . 5 (∃𝑗 ∈ ℕ0s 𝑖 = (𝑗 +s 1s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
586, 57syl 17 . . . 4 ((𝑖 ∈ ℕ0s𝑖 ≠ 0s ) → 𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
591, 58sylbi 217 . . 3 (𝑖 ∈ ℕs𝑖 ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
6059ssriv 3986 . 2 s ⊆ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
61 fveq2 6905 . . . . . . 7 (𝑘 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅))
6261eleq1d 2825 . . . . . 6 (𝑘 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs))
63 fveq2 6905 . . . . . . 7 (𝑘 = 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗))
6463eleq1d 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 6905 . . . . . . 7 (𝑘 = suc 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗))
6665eleq1d 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 6905 . . . . . . 7 (𝑘 = 𝑖 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖))
6867eleq1d 2825 . . . . . 6 (𝑘 = 𝑖 → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs))
6929, 27eqeltri 2836 . . . . . 6 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs
70 peano2nns 28354 . . . . . . 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 7465 . . . . . . . . 9 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V
72 oveq1 7439 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
73 oveq1 7439 . . . . . . . . . 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 ))
7436, 72, 73frsucmpt2 8481 . . . . . . . . 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 ))
7571, 74mpan2 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 ))
7675eleq1d 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))
7770, 76imbitrrid 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))
7862, 64, 66, 68, 69, 77finds 7919 . . . . 5 (𝑖 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs)
7978rgen 3062 . . . 4 𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs
80 fnfvrnss 7140 . . . 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)
8149, 79, 80mp2an 692 . . 3 ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) ⊆ ℕs
8253, 81eqsstri 4029 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ⊆ ℕs
8360, 82eqssi 3999 1 s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  wss 3950  c0 4332  cmpt 5224  ran crn 5685  cres 5686  cima 5687  suc csuc 6385   Fn wfn 6555  cfv 6560  (class class class)co 7432  ωcom 7888  reccrdg 8450   No csur 27685   0s c0s 27868   1s c1s 27869   +s cadds 27993  0scnn0s 28319  scnns 28320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-1s 27871  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec2 27983  df-adds 27994  df-n0s 28321  df-nns 28322
This theorem is referenced by:  nnsind  28364  expsp1  28413
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