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Theorem dfnns2 28377
Description: Alternate definition of the positive surreal integers. Compare df-nn 12265. (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 28358 . . . 4 (𝑖 ∈ ℕs ↔ (𝑖 ∈ ℕ0s𝑖 ≠ 0s ))
2 df-ne 2939 . . . . . . 7 (𝑖 ≠ 0s ↔ ¬ 𝑖 = 0s )
3 n0s0suc 28360 . . . . . . . 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 7438 . . . . . . . . . . . . 13 (𝑖 = 0s → (𝑖 +s 1s ) = ( 0s +s 1s ))
8 1sno 27887 . . . . . . . . . . . . . 14 1s No
9 addslid 28016 . . . . . . . . . . . . . 14 ( 1s No → ( 0s +s 1s ) = 1s )
108, 9ax-mp 5 . . . . . . . . . . . . 13 ( 0s +s 1s ) = 1s
117, 10eqtrdi 2791 . . . . . . . . . . . 12 (𝑖 = 0s → (𝑖 +s 1s ) = 1s )
1211eqeq2d 2746 . . . . . . . . . . 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 3177 . . . . . . . . . 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 7438 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑖 +s 1s ) = (𝑘 +s 1s ))
1514eqeq2d 2746 . . . . . . . . . . 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 3177 . . . . . . . . . 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 7438 . . . . . . . . . . . . 13 (𝑖 = (𝑘 +s 1s ) → (𝑖 +s 1s ) = ((𝑘 +s 1s ) +s 1s ))
1817eqeq2d 2746 . . . . . . . . . . . 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 3177 . . . . . . . . . . 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 6916 . . . . . . . . . . . 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 3236 . . . . . . . . . . 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 7438 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑖 +s 1s ) = (𝑗 +s 1s ))
2423eqeq2d 2746 . . . . . . . . . . 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 3177 . . . . . . . . . 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 28367 . . . . . . . . . . . 12 1s ∈ ℕs
28 fr0g 8475 . . . . . . . . . . . 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 6916 . . . . . . . . . . . 12 (𝑦 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = 1s ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) = 1s ))
3130rspcev 3622 . . . . . . . . . . 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 6916 . . . . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . 15 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) +s 1s ) ∈ V
36 eqid 2735 . . . . . . . . . . . . . . . 16 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
37 oveq1 7438 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (𝑧 +s 1s ) = (𝑥 +s 1s ))
38 oveq1 7438 . . . . . . . . . . . . . . . 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 8479 . . . . . . . . . . . . . . 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 3625 . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . 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 3177 . . . . . . . . . . . 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 3153 . . . . . . . . . 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 28352 . . . . . . . . 9 (𝑗 ∈ ℕ0s → ∃𝑦 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑦) = (𝑗 +s 1s ))
49 frfnom 8474 . . . . . . . . . 10 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω) Fn ω
50 fvelrnb 6969 . . . . . . . . . 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 5702 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)
5452, 53eleqtrrdi 2850 . . . . . . 7 (𝑗 ∈ ℕ0s → (𝑗 +s 1s ) ∈ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω))
55 eleq1 2827 . . . . . . 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 3146 . . . . 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 3999 . 2 s ⊆ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω)
61 fveq2 6907 . . . . . . 7 (𝑘 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅))
6261eleq1d 2824 . . . . . 6 (𝑘 = ∅ → (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) ∈ ℕs ↔ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs))
63 fveq2 6907 . . . . . . 7 (𝑘 = 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗))
6463eleq1d 2824 . . . . . 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 6907 . . . . . . 7 (𝑘 = suc 𝑗 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘suc 𝑗))
6665eleq1d 2824 . . . . . 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 6907 . . . . . . 7 (𝑘 = 𝑖 → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑘) = ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖))
6867eleq1d 2824 . . . . . 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 2835 . . . . . 6 ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘∅) ∈ ℕs
70 peano2nns 28368 . . . . . . 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 7464 . . . . . . . . 9 (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑗) +s 1s ) ∈ V
72 oveq1 7438 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝑦 +s 1s ) = (𝑥 +s 1s ))
73 oveq1 7438 . . . . . . . . . 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 8479 . . . . . . . . 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 2824 . . . . . . 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 3061 . . . 4 𝑖 ∈ ω ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) ↾ ω)‘𝑖) ∈ ℕs
80 fnfvrnss 7141 . . . 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 4030 . 2 (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 1s ) “ ω) ⊆ ℕs
8360, 82eqssi 4012 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 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  cmpt 5231  ran crn 5690  cres 5691  cima 5692  suc csuc 6388   Fn wfn 6558  cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448   No csur 27699   0s c0s 27882   1s c1s 27883   +s cadds 28007  0scnn0s 28333  scnns 28334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008  df-n0s 28335  df-nns 28336
This theorem is referenced by:  nnsind  28378  expsp1  28427
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