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Definition df-nninf 6781
Description: Define the set of nonincreasing sequences in 2𝑜𝑚 ω. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as 0* as defined at df-xnn0 8727 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used ω or 0, but the former allows us to take advantage of 2𝑜 = {∅, 1𝑜} (df2o3 6187) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)
Assertion
Ref Expression
df-nninf = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
Distinct variable group:   𝑓,𝑖

Detailed syntax breakdown of Definition df-nninf
StepHypRef Expression
1 xnninf 6779 . 2 class
2 vi . . . . . . . 8 setvar 𝑖
32cv 1288 . . . . . . 7 class 𝑖
43csuc 4190 . . . . . 6 class suc 𝑖
5 vf . . . . . . 7 setvar 𝑓
65cv 1288 . . . . . 6 class 𝑓
74, 6cfv 5010 . . . . 5 class (𝑓‘suc 𝑖)
83, 6cfv 5010 . . . . 5 class (𝑓𝑖)
97, 8wss 2999 . . . 4 wff (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)
10 com 4403 . . . 4 class ω
119, 2, 10wral 2359 . . 3 wff 𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)
12 c2o 6167 . . . 4 class 2𝑜
13 cmap 6395 . . . 4 class 𝑚
1412, 10, 13co 5644 . . 3 class (2𝑜𝑚 ω)
1511, 5, 14crab 2363 . 2 class {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
161, 15wceq 1289 1 wff = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
Colors of variables: wff set class
This definition is referenced by:  infnninf  6795  nnnninf  6796  0nninf  11776  nninff  11777  nnsf  11778  peano4nninf  11779  nninfalllemn  11781  nninfalllem1  11782  nninfex  11784  nninfself  11788
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